mathematical physics, analysis and geometry - volume 5

Mathematical Physics, Analysis and Geometry 5: 1–63, 2002.© 2002 Kluwer Academic Publishers. Printed in the Netherlands.

1

Inverse Scattering, the Coupling ConstantSpectrum, and the Riemann Hypothesis

N. N. KHURIDepartment of Physics, The Rockefeller University, New York, NY 10021, U.S.A.

(Received: 28 December 2001)

Abstract. It is well known that the s-wave Jost function for a potential, λV , is an entire function ofλ with an infinite number of zeros extending to infinity. For a repulsive V , and at zero energy, thesezeros of the ‘coupling constant’, λ, will all be real and negative, λn(0) < 0. By rescaling λ, such thatλn < −1/4, and changing variables to s, with λ = s(s − 1), it follows that as a function of s theJost function has only zeros on the line sn = 1/2 + iγn. Thus, finding a repulsive V whose couplingconstant spectrum coincides with the Riemann zeros will establish the Riemann hypothesis, but thiswill be a very difficult and unguided search.

In this paper we make a significant enlargement of the class of potentials needed for a generaliza-tion of the above idea. We also make this new class amenable to construction via inverse scatteringmethods. We show that all one needs is a one parameter class of potentials, U(s; x), which areanalytic in the strip, 0 � Re s � 1, Im s > T0, and in addition have an asymptotic expansion inpowers of [s(s − 1)]−1, i.e.

U(s; x) = V0(x) + gV1(x) + g2V2(x) + · · · + O(gN),

with g = [s(s − 1)]−1. The potentials Vn(x) are real and summable. Under suitable conditions onthe V ′

ns and the O(gN) term we show that the condition,∫∞

0 |f0(x)|2V1(x) dx �= 0, where f0 is thezero energy and g = 0 Jost function for U , is sufficient to guarantee that the zeros gn are real and,hence, sn = 1/2 + iγn, for γn � T0.

Starting with a judiciously chosen Jost function, M(s, k), which is constructed such that M(s, 0)is Riemann’s ξ(s) function, we have used inverse scattering methods to actually construct a U(s; x)with the above properties. By necessity, we had to generalize inverse methods to deal with com-plex potentials and a nonunitary S-matrix. This we have done at least for the special cases underconsideration.

For our specific example,∫∞

0 |f0(x)|2V1(x) dx = 0 and, hence, we get no restriction on Im gnor Re sn. The reasons for the vanishing of the above integral are given, and they give us hints onwhat one needs to proceed further. The problem of dealing with small but nonzero energies is alsodiscussed.

Mathematics Subject Classifications (2000): 81U40, 11M26, 11M06, 81U05.

Key words: Riemann hypothesis, inverse scattering.

2 N. N. KHURI

1. Introduction

Many physicists have been intrigued by the Riemann conjecture on the zeros of thezeta function. The main reason for this is the realization that the validity of the hy-pothesis could be established if one finds a self-adjoint operator whose eigenvaluesare the imaginary parts of the nontrivial zeros. The hope is that this operator couldbe the Hamiltonian for some quantum mechanical system. Results by Dyson [1],and Montgomery [2] first made the situation more promising. The pair distributionbetween neighboring zeros seemed to agree with that obtained for the eigenvaluesof a large random Hermitian matrix. But later numerical work showed correlationsbetween distant spacings do not agree with those of a random Hermitian matrix.The search for such a Hamiltonian in physical problems has eluded all efforts.Berry [3] has suggested the desired Hamiltonian could result from quantizing somechaotic system without time reversal symmetry. This seems to be in better agree-ment with numerical work on the correlations of the Riemann zeros, but one is stillfar from even a model or example. It is useful to explore new ideas.

Our choice for this paper is an idea originating from Chadan [4]. In this ap-proach, one tries to relate the zeros of the Riemann zeta function to the ‘couplingconstant spectrum’ of the zero energy, S-wave, scattering problem for repulsivepotentials. We sketch this idea briefly.

The Schrödinger equation on x ∈ [0,∞) is

−d2f

dx2(λ; k; x) + λV (x)f (λ; k; x) = k2f (λ; k; x), (1.1)

where k is the wave number, λ a parameter physicists call the coupling constant,V (x) is a real potential satisfying an integrability condition as in Equation (2.2)below, and f is the Jost solution determined by a boundary condition at infinity,(e−ikxf ) → 1 as x → +∞. The Jost function, M(λ; k), is defined bylimx→0 f (λ; k; x) = M(λ; k). It is well known that M is also the Fredholm deter-minant of the Lippmann–Schwinger scattering integral equation for S-waves. Bothf (λ; k; x) and M(λ; k) are, for any fixed x � 0, analytic in the product of the halfplane, Im k > 0, and any large bounded region in the λ plane. In fact, it is knownthat for any fixed k, Im k � 0, M(λ; k) is entire in λ and of finite order. ThusM(λ; k) has an infinite number of zeros, λn(k), with λn(k) → ∞ as n → ∞.

Starting with Equation (1.1), and its complex conjugate with k = iτ , τ > 0,and setting λ = λn(iτ ), we obtain

[Im λn(iτ )]∫ ∞

0|f (λn(iτ ); iτ ; x)|2V (x) dx = 0. (1.2)

For the class of potentials, we deal with V = O(e−mx) as x → ∞. Thus, we cantake the limit τ → 0, and get

[Im λn(0)]∫ ∞

0|f (λn(0); 0; x)|2V (x) dx = 0. (1.3)

INVERSE SCATTERING AND THE RIEMANN HYPOTHESIS 3

Hence, for repulsive potentials, V (x) � 0, all the zeros λn(0) are real. For anyτ, τ > 0, the same is true for all λn(iτ ). But λn(iτ ) must be negative, sincethe potential [λn(iτ )V ] will have a bound state at E = −τ 2, and that could nothappen if V � 0 and λn(iτ ) > 0. Hence, by continuity, λn(0), for all n, is realand negative [5]. The zero energy coupling constant spectrum, λn(0), lies on thenegative real line for V � 0. Chadan’s idea is very simple. He introduces a newvariable, s, and defines

λ ≡ s(s − 1). (1.4)

Thus, one can write

M(λ, 0) = M(s(s − 1); 0) ≡ χ(s). (1.5)

It is easy to see now that, for | Im s| > 1, the zeroes, sn, of χ(s) are all such that

sn = 12 + iγn; λn(0) ≡ sn(sn − 1). (1.6)

The problem is actually somewhat simplified by noting that first we do not needthe condition λn < 0 as long as we restrict ourselves to the strip 0 � Re s � 1, andIm s > 1. Second, it is sufficient to prove that the integral in Equation (1.3) doesnot vanish. Thus, one does not need a fully repulsive potential for the Riemannproblem.

One might comment that it is very difficult to find a potential with

λn(0) = sn(sn − 1) and sn = 12 ± iγn,

sn being the Riemann zeros. But it is probably as difficult as finding an Hermitianoperator whose eigenvalues are γn. Indeed, the latter may be impossible withoutintroducing chaotic systems.

The results mentioned above also apply when V = V0 +λV1, with only V1 � 0,and V0, V1 both real and satisfying Equation (2.2) and with certain restrictions onV0. This remark leads directly to the basic idea of this paper the of objective ofwhich is to show that the coupling constant approach can be significantly simplifiedand made amenable to inverse scattering methods.

Our first remark is that one does not need a potential, V = V0 +λV1, dependinglinearly on the coupling parameter λ. Given a one-parameter family of complexpotentials, U(s; x), x ∈ [0,∞), which for fixed x are analytic in s in the strip,0 � Re s � 1, Im s > T0 > 2, we can, following similar arguments as above,obtain, for s = sn, sn being a zero of the zero energy Jost function,∫

|f (sn; 0; x)|2[ImU(sn; x)] dx ≡ 0, (1.7)

where f is the zero energy Jost solution evaluated at s = sn.Next, suppose in addition to the above properties, U has an asymptotic expan-

sion in inverse powers of s, actually better, s(s − 1), i.e.

U(s, x) = V0(x) + gV1(x) + g2V2(x) + · · · + gNV(N)R (g; x), (1.8)

4 N. N. KHURI

where

g ≡ 1

s(s − 1). (1.9)

Under suitable conditions on the Vn(x) and estimates of the O(gN) term, and itsphase, one again gets

[Im gn]∫ ∞

0|f (0; 0; x)|2V1(x) dx = 0, (1.10)

with gn = [sn(sn − 1)]−1, the sn’s are the zeros of M(s, 0) the zero energy Jostfunction, and f (g; k; x) is the Jost solution with the full U . The result (1.10) isonly established for zeros with Im sn > T0, where T0 is large enough for the V1

contribution to Equation (1.7) to dominate the integral in Equation (1.7). However,this is sufficient, since the Riemann hypothesis has already been proved for zeroswith | Im sn| < T , where T could be as large as 105.

Again, all we need for sn = 1/2 + iγn is to have the integral in (1.10) notvanishing. In the end, only the properties of V1 matter.

In this paper we will use inverse scattering methods, albeit for complex po-tentials, to actually prove the existence of such a U(s; x). By construction, thispotential has the additional property that the zero energy Jost function is Riemann’sξ function,

limk→0

M(s; k) ≡ 2ξ(s). (1.11)

We will also give explicit expressions for V0, V1, V2, and bounds on V(N)R .

The difficult point turns out to be that, in our specific example,∫ ∞

0|f (0; 0; x)|2V1(x) dx ≡ 0. (1.12)

Thus, we get no information on [Im gn], or Re(sn − 1/2). We shall discuss whatone needs to proceed further. This will require working with small, but nonzeroenergy values.

We start by introducing a special class of Jost functions, M±, which depend onan extra parameter ν = s − 1/2, with the property that the zero energy limit,

limk→0

M±(ν, k) = 2ξ(ν + 12 ).

For fixed ν, the Jost functions are taken to be of the Martin [8] type, i.e. having cutplane analyticity in the momentum variable k. This is the class of Jost functionsthat results when the potential is a superposition of Yukawa potentials. We then useinverse scattering methods to prove the existence of a complex potential U(ν, τ)

which is determined uniquely by the initial S-matrix. We do carry out the analysisfor ν in the truncated critical strip, i.e. −1/2 < Re ν < 1/2, and Im ν > T0, with


T0 > 16π2. This is, of course, the domain most relevant to the Riemann problem.Standard techniques of inverse scattering are not immediately applicable, becauseS(ν, k) does not satisfy the reality condition, and is not unitary for complex ν.However, we shall see that, in our specific case, we can bypass these difficultiesand carry out an inverse scattering procedure anyway. We have attempted to makethe paper self-contained and do not rely on results that need the unitarity of S inthe proof.

In Section 2, we give a brief review of relevant scattering theory results in-tended for mathematicians not familiar with them. This review also helps defineour physics terminology.

Section 3 is devoted to the introduction of our special class of Jost functions,M±(ν; k). Following that, in Section 4, we briefly discuss the real ν case, whichis a standard inverse scattering case covered by well-known results. This section isinstructive, even though real ν is uninteresting for the Riemann problem. The nextstep, Section 5, is to study in more detail the properties of M±. The main result isan asymptotic expansion in powers of a variable, g ≡ (ν2 − 1/4)−1, which gives

M(±) = M(±)

0 (k) + gM(±)

1 (k) + g2M(±)

2 + · · · + gNR(±)N (g; k). (1.13)

Here all the M(±)n can be computed exactly via recursion formulae, and in addition,

for real k, they satisfy

[M(+)n (k)]∗ = M(−)

n (k) and M(+)n (−k) = M(−)

n (k).

The remainder functions, R(±)N , are given explicitly and are O(g) as g → 0.

In Section 6, with fixed ν in the strip, we determine the number and positionsof zeros in the upper half k-plane. It turns out that there is at most one such zeroand it lies close to the origin. In fact we can give a good estimate of its position.

Section 7 is devoted to the study of the case |ν| → ∞, i.e. |g| → 0. HereM(±)(ν, k) → M

(±)0 (k) which is a known rational function in k. This leads to

an exactly soluble Marchenko equation and an exact result for the correspondingV0(x).

Section 8 is devoted to proving the existence of solutions of the Marchenkoequation for our specific class of S-matrices. With the resulting Marchenko op-erator, A(ν; x, y), which is now complex, we proceed to define in the standardway a potential U(ν, x) and corresponding Jost solutions, f (±)(ν; k, x), of theSchrödinger equation. Finally, we check directly that indeed f (±) are solutionsof the Schrödinger equation with the desired asymptotic properties. The main dif-ference from the standard case is that U(ν; x) is now complex unless ν is purelyimaginary.

In Section 9 we discuss the case ν = it , t real. This is a standard inverse problemwith S(it, k) unitary for k real, and the resulting U(it; x) is real.

More detailed properties of V (ν, x) are given in Section 10. There we give anasymptotic expansion,

V (ν, x) = V0(x) + gV1(x) + g2V2(x) + · · · + gNV(N)R (g, x)

6 N. N. KHURI

with all Vn’s real and all representable by superpositions of Yukawa potentials.Also Vn(x) is continuous and differentiable for x ∈ [0,∞), and Vn(0) is finite. Forcompleteness we calculate V1(x) explicitly, and indicate how Vn(x), n > 1, caneasily be computed. We also give some needed properties of V (N)

R and of [ImV(N)R ]

for small Re ν.In Section 11, we study the zeros, νn(k), of M−(ν, k) for small fixed k with

Im k � 0. We prove that νn(0) are the standard Riemann zeros, and also that|νn(k) − νn(0)| = O(k1/p) for small k. Here p is the multiplicity of the Riemannzero νn = νn(0). We also prove that any Riemann zero, νj , is the limit of a zero ofM(−)(ν; k), νj (k), as k → 0.

Finally, in Section 12 we discuss the relation of our potential, V (g; x), and itsJost solutions to the Riemann hypothesis. We prove that in this case∫ ∞

0|f (0; 0; x)|2V1(x) dx = 0

and, hence, no information on the Riemann hypothesis can result directly fromthis example at zero energy. But the reasons for the failure are clear, and theyindicate the properties of a desired Jost function that will be sufficient to makethe important step. The fact that one can set k = iτ , τ > 0 but small, and try toprove the hypothesis for νn(τ), τ arbitrarily small, but τ �= 0, provides a significantsimplification of the problem.

2. A Sketch of Scattering Theory

This section is intended to facilitate the reading of this paper by those mathe-maticians (or physicists) who are not familiar with elementary scattering theoryin quantum mechanics. At the end of this section we will give a list of books andreview papers where more information can be obtained.

The Schrödinger equation for s-waves is given by

−d2f

dx2+ gV (x)f = k2f, k = κ + iτ. (2.1)

Here x ∈ [0,∞), V (x) is real, g is a parameter that physicists call a couplingconstant. The reason for introducing it will become apparent below. One studiesthe class of real potentials, V (x), which are locally summable functions and satisfythe condition,∫ ∞

0x|V (x)|eαx dx = C < ∞, 0 � α � m. (2.2)

For scattering theory, the important solutions of Equation (2.1) are the so-calledJost solutions [9]. These are the two linearly independent solutions, f (±)(g, k, x)

with boundary values at infinity given by

limx→∞ e±ikxf (±) = 1. (2.3)


Using the method of variation of parameters, we can replace Equation (2.1) andthe condition (2.3) by an integral equation:

f (±)(g; k; x) = e∓ikx + g

∫ ∞

x

sin k(x′ − x)

kV (x′)f (±)(g; k; x′) dx′. (2.4)

Starting with the papers of Jost [9] and Levinson [10], the existence of solu-tions to Equation (2.4) and their properties have been well established for V (x)

satisfying the condition (2.2).The basic input needed is the upper bound on the kernel,∣∣∣∣sin k(x′ − x)

k(x′ − x)

∣∣∣∣ � C1e|τ ||x ′−x|

1 + |k||x′ − x| , Im k ≡ τ, (2.5)

where C1 is O(1). With this bound and the bound (2.2) one proves the absoluteconvergence of the iterative series of the Volterra equation (2.4) for any x � 0,and k with Im k > −(m/2) for f (−), and Im k < m/2 for f (+). Also it is easyto prove that, for any finite g and x � 0, f (+)(g; k; x) is an analytic functionof k for Im k < m/2. Similarly, f (−)(g; k; x) is analytic in Im k > −(m/2). Inaddition, for k in the analyticity domain, the power series in g obtained by iteratingEquation (2.4) is absolutely and uniformly convergent for g inside any finite regionin the g-plane. Thus both f ±(g; k; x) are entire functions of g.

The scattering information is all contained in the Jost functions, denoted byM(±)(g; k) and defined by

M(±)(g; k) ≡ limx→0

f (±)(g; k; x). (2.6)

Both limits in Equation (2.6) exist for finite |g|, and k in the respective domain ofanalyticity, for all potentials satisfying the condition (2.2). The S-matrix is givenby

S(g; k) ≡ M(+)(g; k)M(−)(g; k) . (2.7)

For real g and Im k > 0, M(−)(g, k) has no zeros except for at most a finite numberon the imaginary k-axis. These zeros, kn = iτn, give the point spectrum of theHamiltonian of (2.1) with En = −τ 2

n . Their number cannot exceed the value of theintegral

∫∞0 x|V | dx, a result due to Bargmann [11].

Another important property of M(−)(g; k) was first obtained by Jost and Pais [12].The regular solution of Equation (2.1), φ(g; k; x), with φ(g; k; 0) = 0, is

φ(g; k; x) ≡ 1

2ik[M(+)(g; k)f (−)(g; k; x) − M(−)(g; k)f (+)(g; k; x)]. (2.8)

The solution φ satisfies a Fredholm type integral equation which, for poten-tials satisfying (2.2), was studied in [12]. Jost and Pais demonstrate explicitly that

8 N. N. KHURI

M(−)(g, k) is identical to the Fredholm determinant of the scattering integral equa-tion for φ. Hence, for any fixed k, with Im k > 0, the zeros of M(−)(g; k) in theg-plane, gn(k), give the ‘coupling constant eigenvalues’ at which the homogeneousFredholm equation has solutions, φ = gn(k)Kφ. Since, M(−)(g, k) is an entirefunction of finite order in g, the sequence g1(k), g2(k), . . . , gn(k) tends to infinityas n → ∞.

For the purposes of this paper a result of Meetz [5] is instructive. Let us con-sider a potential which is repulsive, i.e. V > 0 for all x ∈ [0,∞). Then fork = iτ, τ > 0, the coupling constant spectrum, gn(iτ ), is real and negative. Thisresult is implicitly contained in [12].

In this brief review we need to make an important remark about complex po-tentials, V �= V ∗. Mathematicians and mathematical physicists often ignore thesepotentials. The Hamiltonian is no longer self-adjoint if V �= V ∗, with g = 1.But physicists, especially those who work on nuclear physics, do not have such aluxury. There are many interesting and useful models, especially in nuclear physics,where V is complex. Of course, the general and beautiful results which hold forreal V do not all apply for complex V . But many survive, and one has just to becareful which to use and to establish alternative ones when needed.

There are many books that cover inverse scattering. But for the purposes ofthis paper, we recommend the book of Chadan and Sabatier [13], since it alsodiscusses the superposition of the Yukawa case and the Martin results. For thestandard results on inverse scattering, the review paper by Faddeev [14] is highlyrecommended.

3. A Special Class of Jost Functions

In this section we will combine two results whose progeny could not be moredifferent to obtain a representation for a class of Jost functions that we shall studyin detail. The first is Martin’s representation for the Jost functions of the class ofpotentials that can be represented as a Laplace transform. The second is Riemann’sformula for the function ξ(s) defined below.

Starting 40 years ago, physicists [15, 16], for reasons not relevant to this paper,studied the class of potentials that, in addition to satisfying Equation (2.2), have aLaplace transform representation, i.e. for all x > 0,

V (x) =∫ ∞

m

C(α)e−αx dα, m > 0, (3.1)

where C(α) is summable and restricted to satisfy∫∞m

|C(α)|α−2 dα < ∞. This lastcondition guarantees that x|V (x)| is integrable at x = 0.


For these potentials Martin) proved that the Jost functions M(±)(k) have therepresentation

M(±)(k) = 1 +∫ ∞

m2

w(α)

α ± ikdα. (3.2)

Here w is real and summable and is such that M(±) → 1 as |k| → ∞. We haveset g = 1 here. Note that not any arbitrarily chosen summable w(α) is accept-able. M(−)(k) must have no zeros for Im k > 0 except for a finite number on theimaginary k-axis corresponding to the point spectrum.

For our purposes, here we choose a specific family of functions M(±)(ν; k)defined such that

M(±)(ν; 0) ≡ 2ξ(ν + 12 ), (3.3)

where

ξ(s) = 12s(s − 1)π−s/2+

(s

2

)ζ(s) (3.4)

and

ζ(s) =∞∑n=1

n−s , Re s > 1. (3.5)

Riemann’s formula for ξ(s) defines an entire function of order one in s, and isgiven by [18]

2ξ(s) = 1 + s(s − 1)∫ ∞

1ψ(α)[αs/2−1 + α−1/2−s/2] dα, (3.6)

where

ψ(α) =∞∑n=1

e−πn2α, α � 1. (3.7)

We also have the symmetry relation ξ(s) = ξ(1 − s).For convenience we define the variable, ν, as

s ≡ 12 + ν. (3.8)

With this variable ξ(1/2 + ν) is symmetric in ν, and we have

2ξ( 12 + ν) = 1 + (ν2 − 1

4 )

∫ ∞

1ψ(α)α−3/4[αν/2 + α−ν/2] dα. (3.9)

) See [8] and [17].

10 N. N. KHURI

Our starting point is to define two functions, M±(ν; k), as

M(±)(ν; k) ≡ 1 + (ν2 − 14 )

∫ ∞

1

ψ(α)α1/4[αν/2 + α−ν/2]α ± ik

dα. (3.10)

This definition holds for any finite, real or complex, ν, and for any k excluding thecuts on the imaginary k-axis, k = iτ , 1 � τ < ∞, for M(+), and −1 � τ > −∞,for M(−).

Obviously, we have

M(±)(ν; 0) ≡ 2ξ( 12 + ν). (3.11)

In addition, the fact that ψ(α) = O(e−πα) as α → +∞, guarantees that for anyfinite |ν|,

lim|k|→∞M±(ν; k) = 1. (3.12)

This is true along any direction in the complex k-plane excluding the pure imag-inary lines. But even for arg k = ±π/2, the limit holds using standard results.The immediate question that faces us at this stage is: for which regions in the ν-plane, if any, can one use the functions M(±)(ν; k) defined in Equation (3.10) asJost functions and proceed to use the resulting S-matrix, S(ν; k), as the input in aninverse scattering program.

There are two issues involved. The first, and most important, is to make surethat M(−)(ν; k), has no complex zeros in k for Im k > 0, except for a finite numberon the imaginary axis. This is not true for any ν. But fortunately for the set of ν’smost important to the Riemann hypothesis, M(−)(ν; k) has at most one zero closeto the origin with Im k > 0. This will be shown in Section 6.

The second issue relates to the question of reality. For real potentials V and realk, we have the relations

[M(+)(k)]∗ = M(−)(k) and |S(k)| = 1.

Clearly, for complex ν, this does not hold for M(±)(ν; k). However, we will provethat for those values of ν in the truncated critical strip, one can still carry out theinverse scattering program and obtain a unique and well-defined V (ν; x) which, ofcourse, could now be complex. Since the old results of inverse scattering theory alluse the fact that, |S(k)| = 1, we have to go back to square one and prove everystep anew for the present case. Our task is tremendously simplified by the fact that,even though S(ν; k) is not unitary, we still have |S(ν; k)| = 1 + O(1/|ν|2), and weare only interested in |ν| > 103.


4. The Real ν Case

For potentials satisfying the representation (3.1), Martin), in addition to the resultssummarized in Equation (3.2), developed an iterative scheme which enables oneto reconstruct the measure, C(α) in Equation (3.1) from the knowledge of thediscontinuity of S(k) along the branch cut on the imaginary axis,

k = iτ,m

2� τ < ∞.

This gives an inverse scattering method that, at first sight, looks quite differentfrom the standard ones of Gelfand and Levitan [6] and Marchenko [7]. The rela-tion between these two methods was first clarified by Gross and Kayser [19] andindependently by Cornille [20]. They showed that for potentials of the form (3.1)the Marchenko kernel is a Laplace transform of the discontinuity of S(k), and theycarried out an extensive analysis of the relation between Martin’s and Marchenko’smethods. These results were reviewed and enlarged in a more recent paper by theauthor [21].

For ν real and |ν| > 1/2, the functions M(±)(ν; k) defined in Equation (3.10)are indeed bona fide Jost functions with (M(+)(ν; k))∗ = M(−)(ν; k) for real k.The positivity of ψ(α) guarantees the absence of a point spectrum. The S-matrixis

S(ν; k) ≡ M(+)(ν; k)M(−)(ν; k) . (4.1)

We define the discontinuity D(ν; τ) as

D(ν, τ) = limε→0

[S(ν; iτ + ε) − S(ν; iτ − ε)], τ > 1. (4.2)

From Equation (3.10), one obtains

D(ν, τ) = ω(ν; τ)1 + 1

π

∫∞1

ω(ν,β)

β+τdβ

, (4.3)

with

ω(ν, τ) = π(ν2 − 1/4)ψ(τ)τ 1/4[τ ν/2 + τ−ν/2]. (4.4)

For real ν > 1/2, D(ν, τ) and ω(ν, τ) are positive for all τ � 1. The case withω(ν, τ) � 0 is the easiest to handle by the Martin inverse method, and it can bedone explicitly.

Although having ν real and ν > 1/2 is of little direct interest to the Riemannproblem, we give the results here as they might be helpful to the reader. For details,one should consult [21].

) See [8] and [13].

12 N. N. KHURI

The S-matrix, S(ν, k), uniquely determines a potential, V (ν; x), and its Jostsolutions f (±)(ν; k; x). V is given by

V (x) = 4∞∑n=0

(−1

π

)n+1 ∫ ∞

1dα0 . . . ×

×∫ ∞

1dαn

∏nj=0 D(ν;αj)e−2αj x∏n−1

j=0(αj + αj+1)

(n∑

j=0

αj

). (4.5)

This series for V is absolutely and uniformly convergent for all x � 0, and ν >

1/2. This follows from the positivity in (4.3),

1

π

∫ ∞

1

|D(ν, α)|α + τ

dα �1π

∫∞1 ω(ν, α)/α dα

1 + 1π

∫∞1

ω(ν,β)

βdβ

< 1. (4.6)

The Jost solutions, f (±)(ν; k; x) are given by

f (±) = e∓ikx + e∓ikx

∞∑n=0

(−1

π

)n+1 ∫ ∞

1dα0 . . .

∫ ∞

1dαn ×

×∏n

j=0 D(ν;αj)e−2αj x

[∏n−1j=0(αj + αj+1)][α0 ± ik] . (4.7)

Again this last series is absolutely and uniformly convergent for all x � 0, and k

in a compact domain inside the respective regions of analyticity.One can check directly that f ± given by Equation (4.6) are solutions of the

Schrödinger equation with V (ν; x) of Equation (4.4) as potential, see [21] for moredetails.

5. Some Properties of M(±)(ν, k) for |ν| > 1

To proceed further and study M(±)(ν, k) for complex ν, and more specifically ν inthe critical strip, −1/2 < Re ν < 1/2; Im ν > 1, the defining representation (3.10)is not fully instructive. This is because the behavior of M± for large Im ν is notadequately shown by Equation (3.10). Our final result in this section is to obtain anasymptotic expansion of M±(ν, k) for fixed k in inverse powers of (ν2 − 1/4).

We need to carry out integrations by parts on the integrand in Equation (3.10)analogous to those performed in Titchmarsh’s book [18], for Equation (3.9).

The following lemma will prove extremely useful:

LEMMA 5.1. Let W(α), α ∈ [1,∞), be a C∞ function, and W(α) = O(e−α) asα → ∞, then given the integral

I (ν) =∫ ∞

1W(α)[αν/2 + α−ν/2] dα, (5.1)


one has, after two integrations by parts,

I (ν) = 1

ν2 − 14

{8[W(1) + (W ′(α))α=1] +

+∫ ∞

1W1(α)[αν/2 + α−ν/2] dα

}, (5.2)

where

W1(α) = 154 W(α) + 12αW ′(α) + 4α2W ′′(α). (5.3)

Proof. We rewrite Equation (5.1) as

I (ν) =∫ ∞

1dα(W(α)α3/4)[αν/2−3/4 + α−ν/2−3/4]. (5.4)

Integrating by parts, we get

I (ν) = 2W(1)

ν2 − 14

−∫ ∞

1dα

(d

dα[W(α)α3/4]

)[αν/2+1/4

ν2 + 1

4

− α−ν/2+1/4

ν2 − 1

4

]. (5.5)

This again can be rewritten as

I (ν) = 2W(1)

ν2 − 14

−∫ ∞

1dα

{α3/2

(d

dα[W(α)α3/4]

)}[αν/2−5/4

ν2 + 1

4

− α−ν/2−5/4

ν2 − 1

4

]. (5.6)

Carrying out a second integration by parts, we obtain

I (ν) = 1

ν2 − 14

[2W(1) + 8

{d

dα(W(α)α3/4)

}α=1

]+

+ 4

(ν2 − 14 )

∫ ∞

1dαα−1/4

{d

dα

(α3/2

[d

dα(W(α)α3/4)

])}×

×[αν/2 + α−ν/2]. (5.7)

Performing the differentiations in (5.7) easily leads to Equation (5.2). ✷We can apply this lemma to the integral in Equation (3.10) which defines

M(±)(ν, k). Setting

W(±)(α; k) ≡ ψ(α)α1/4

α ± ik, (5.8)

and restricting k to the corresponding domain of analyticity in k

P (+) = {k | Im k < 1}; P (−) = {k | Im k > −1}, (5.9)

14 N. N. KHURI

we get

M(±)(ν; k) = M(±)0 (k) +

∫ ∞

1dαW(±)

1 (α; k)[αν/2 + α−ν/2], (5.10)

with

W(±)

1 =3∑

5=1

6(1)5 (α)

(α ± ik)5, (5.11)

and

6(1)1 (α) = 6ψ(α)α1/4 + 14ψ ′(α)α5/4 + 4ψ ′′(α)α9/4,

6(1)2 (α) = −14ψ(α)α5/4 − 8ψ ′(α)α9/4,

6(1)3 (α) = 8ψ(α)α9/4. (5.12)

The first term in (5.10) is independent of ν, and given by

M(±)

0 (k) = 1 + a1

1 ± ik+ a2

(1 ± ik)2, (5.13)

with

a1 = −1 + 8ψ(1); a2 = −8ψ(1). (5.14)

In obtaining (5.14), we have used the identity

4ψ ′(1) + ψ(1) = − 12 . (5.15)

It is important to note that both a1 and a2 are negative and that a1 + a2 = −1. Thisleads to

limk→0

M(±)

0 (k) = 0. (5.16)

As a check on Equation (5.10), we take the k → 0 limit

M(±)(ν; 0) =∫ ∞

1dα

[3∑

5=1

(6(1)5 /α5)

][αν/2 + α−ν/2]. (5.17)

Substituting the expressions for 6(1)5 given in Equation (5.12), we get

M(±)(ν, 0) = 4∫ ∞

1dα(ψ ′′(α)α5/4 + 3

2ψ′(α)α1/4)[αν/2 + α−ν/2]

= 2ξ( 12 + ν). (5.18)

Lemma 5.1 can be used repeatedly to give an asymptotic expansion of M±(ν; k)in inverse powers of (ν2 −1/4). Recursion formulae can be given to give each termfrom the preceding one.


Indeed given

I (±)n (ν; k) = 1

(ν2 − 14 )

n−1

∫ ∞

1dαW(±)

n (α; k)[αν/2 + α−ν/2], (5.19)

with

W(±)n =

2n+1∑5=1

6(n)5 (α)

[α ± ik]5 , (5.20)

and k ∈ P±, one obtains

I(±)

n+1 = 1

(ν2 − 14)

n

{8

[W(±)

n (1; k) +(

dW(±)n

dα

)α=1

]+

+∫ ∞

1dαW(±)

n+1(α; k)[αν/2 + α−ν/2]}, (5.21)

and

W(±)

n+1 = 154 W

(±)n + 12α(W(±)

n )′ + 4α2(W(±)n )′′, (5.22)

where the primes indicate differentiation with respect to α. Again W(±)

(n+1) will beas in Equation (5.20)

W±(n+1) =

2n+3∑5=1

6(n+1)5 (α)

[α ± ik]5 . (5.23)

For each n we have 1 � 5 � 2n + 1, and the functions 6(n)5 satisfy a recursion

formula, which follows from (5.22).

6(n+1)5 (α) = 15

4 6(n)5 + 12α(6(n)

5 )′ − 12α(5 − 1)6(n)

5−1 ++ 4α2[(6(n)

5 )′′ − 2(5 − 1)(6(n)

5−1)′ +

+ (5 − l)(5 − 2)6(n)5−2]. (5.24)

All the 6(n)5 can thus be determined by iteration starting from

6(0)1 (α) ≡ ψ(α)α1/4. (5.25)

The general form of 6(n)

5 (α) is easily determined to be

6(n)5 (α) =

2n+1−5∑j=0

C(n)(5; j) α1/4+5+j−1ψ(j)(α). (5.26)

The coefficients C(n)(5; j) are real, and C(0)(1; 0) = 1 determines all the others.Also

ψ(j)(α) ≡(

d

dα

)j

ψ(α). (5.27)

16 N. N. KHURI

At this point we can substitute Equation (5.26) in (5.24) and obtain a recursionformula for C(n)(5; j),

C(n+1)(5, j)

= C(n)(5; j)[ 154 + 12( 1

4 + 5 + j − 1)++ 4( 1

4 + 5 + j − 1)( 14 + 5 + j − 2)]−

−C(n)(5 − 1, j)[12(5 − 1) + 8(5 − 1)( 14 + 5 + j − 2)]+

+C(n)(5, j − 1)[12 + 8( 14 + 5 + j − 2)] + 4C(n)(5, j − 2)−

− 8(5 − l)C(n)(5 − 1; j − 1) + 4(5 − 1)(5 − 2)C(n)(5 − 2; j). (5.28)

Here we have

1 � 5 � 2n + 1, 0 � j � 2n + 1 − 5. (5.29)

For all other values of 5 and j , C(n)(5, j) ≡ 0.Starting with

C(0)(1, 0) ≡ 1, (5.30)

we can compute all other C(n)(5, j). For example, C(1)(1, 0) = 6, C(1)(1, 1) = 14,and C(1)(1, 2) = 4. This agrees with the direct calculation given in Equation (5.12).In Table I, we give all the coefficients C(n)(5, j) up to n = 4. All the coefficientsare integers.

Finally, we give the general form of the surface term in Equation (5.21). Wedefine M(±)

n (k),

M(±)n (k) = 8

[W(±)

n (1, k) +(

dW(±)n

dα

)α=1

]. (5.31)

From Equations (5.20) and (5.26) we obtain, after some algebra,

M(±)n (k) =

2n+2∑5=1

χ(n)5

[1 ± ik]5 , (5.32)

with

χ(n)5 = 8

2n+2−5∑j=0

C(n)(5, j){( 14 + 5 + j)ψ(j)(1) + ψ(j+1)(1)} −

− 82n+2−5∑j=0

(5 − 1)C(n)(5 − 1, j)ψ(j)(1). (5.33)

For the purposes of this paper, it is sufficient to apply our lemma up to the n = 3level. We introduce g, as a new variable:

g ≡ 1

ν2 − 14

. (5.34)


Table I. Values of C(n)(l; j) for n � 3

C(0) j = 0

l = 1 1

C(1) j = 0 1 2

l = 1 6 14 4

2 −14 −8

3 8

C(2) j = 0 1 2 3 4

l = 1 36 364 500 176 16

2 −364 −1000 −528 −64

3 1000 1056 192

4 −1056 −384

5 384

C(3) j = 0 1 2 3 4 5 6

l = 1 216 7784 29152 29128 10448 1440 64

2 −7784 −58304 −87384 −41792 −7200 −384

3 58304 174768 125376 28800 1920

4 −174768 −250752 −86400 −7680

5 250752 172800 23040

6 −172800 −46080

7 46080

Our final result for M(±)(ν, k) with k ∈ P (±) and |ν| � 1,

M(±)(ν; k) = M(±)0 (k) + gM

(±)1 (k) + g2M

(±)2 (k) + g2R

(±)2 (ν, k). (5.35)

Here we have

M(±)

1 (k) =4∑

5=1

b5

[1 ± ik]5 , (5.36)

with b5 ≡ χ(1)5 , and

M(±)2 (k) =

6∑5=1

c5

[1 ± ik]5 , (5.37)

c5 ≡ χ(2)5 . The remainder function R

(±)2 is given by

R(±)2 (ν; k) =

∫ ∞

1

(7∑

5=1

6(3)5 (α)

[α ± ik]5)

[αν/2 + α−ν/2] dα. (5.38)

18 N. N. KHURI

Table II. Values of χ(n)l

for n = 1, 2, 3

χ(n)l

n = 1 n = 2 n = 3

l = 1 10.9973 −460.8231 28967.9828

2 4.7050 −309.0434 36560.3049

3 −7.4045 451.5522 −18626.4002

4 −8.2977 910.6755 −114291.8885

5 71.4580 −76110.8714

6 −663.8194 131495.9200

7 123526.6033

8 −111521.6508

For |Im ν| > 103, the first two terms of Equation (5.35) give a very goodestimate for M(±). We shall explore this in much more detail later. One can goto higher orders in g, but the resulting series is only asymptotic. For our purposes,here Equation (5.35) is enough.

It is important to stress another property of M(±)

1 and M(±)

2 , namely as k → 0,

M(±)

1 (0) = 0, M(±)

2 (0) = 0. (5.39)

We have already shown that M(±)

0 (0) = 0. To check this, we give the explicit formof the coefficients b5 in (5.36). Using χ

(1)5 = b5, Equation (5.33), and Table I, we

get

b1 = 60ψ(1) + 300ψ ′(1) + 216ψ ′′(1) + 32ψ ′′′(1),b2 = −300ψ(1) − 432ψ ′(1) − 96ψ ′′(1),b3 = 432ψ(1) + 192ψ ′(1),b4 = −192ψ(1). (5.40)

Numerically, the b’s are given in Table II.We now have

M(±)

1 (0) =4∑

5=1

b5

= 32[ψ ′′′(1) + 154 ψ

′′(1) + 158 ψ

′(1)]. (5.41)

But

ψ ′′′(1) + 154 ψ

′′(1) + 158 ψ

′(1) = 0. (5.42)

This identity follows from the relation [18]√α(2ψ(α) + 1) = 2ψ(1/α) + 1. (5.43)


Differentiating (5.40) once and setting α = 1 immediately gives Equation (5.15).Differentiating three times leads to Equation (5.42).

Indeed, there is an infinite sequence of identities like Equation (5.42), alwaysstarting with ψ2n+1(1), odd derivatives, which result from differentiating Equa-tion (5.40) (2n + 1) times. Thus, again

M(±)2 (0) =

6∑5=1

c5 = 0, (5.44)

depends on the next identity:

ψ(5)(1) + 454 ψ

(4)(1) + 2354 ψ(3)(1) + 975

8 ψ(2)(1) + 163532 ψ(1)(1) = 0. (5.45)

Of course, only the first two coefficients in (5.45) are unique, since we can alwaysadd a multiple of the left-hand side of (5.42) to (5.45).

The vanishing of M±j (0), j = 0, 1, 2, is indeed necessary since

g2R(±)2 (ν; 0) = g2

∫ ∞

1dα

(7∑

5=1

6(3)5 (α)

α5

)[αν/2 + α−ν/2]

= 2ξ( 12 + ν), (5.46)

which is the result of carrying out four more differentiations by parts on the formulafor ξ(1/2 + ν) given on page 254 of [18].

In Table II, we give the numerical values of χ(n)5 , for n = 1, 2, and 3, and

b5 ≡ χ(1)5 while c5 ≡ χ

(2)5 .

For the convenience we summarize the results of this section:

M(±)(ν, k) =N∑n=0

gnM(±)n (k) + gNR

(±)N (ν, k), (5.47)

where

g = (ν2 − 14)

−1, (5.48)

M(±)n (k) =

2n+2∑5=1

χ(n)5

[1 ± ik]5 , (5.49)

and χ(n)5 are real numbers given in Equation (5.33). In addition, we have

2n+2∑5=1

χ(n)5 ≡ 0, (5.50)

which guarantees that M(±)n (k) → 0 as k → 0. For real k, we have [M(+)

n (k)]∗ =M(−)

n (k).

20 N. N. KHURI

Finally the remainder term R(±)N is given explicitly by

R(±)N (ν, k) =

∫ ∞

1dα

(2N+3∑5=1

6(N+1)5 (α)

(α ± ik)5

)[αν/2 + α−ν/2], (5.51)

with

6(n)5 (α) =

2n+1−5∑j=0

C(n)(5; j)α1/4+5+j−1ψ(j)(α). (5.52)

The C(n)(5, j) are integers determined by a recursion formula given in Equa-tion (5.28), with C(0)(1, 0) ≡ 1, and ψ(j)(α) are the j th derivatives of ψ(α),Equation (5.27).

For k = 0, we have

M(±)(ν, 0) = 2ξ(ν + 12 ). (5.53)

From Equations (5.47) and (5.51) we then have, for any integer n � 0,

2ξ(ν + 12 ) = gn

∫ ∞

1dα

(2n+3∑5=1

6(n+1)5 (α)

α5

)[αν/2 + α−ν/2]. (5.54)

For n = 0, this formula is given in [18], page 225. The results for larger n can beobtained by successive integrations by parts.

6. The Zeroes of M(−)(ν;k) for Im k > 0, and Fixed ν

To study the Riemann hypothesis we need only to focus on the truncated criticalstrip, S(T0),

S(T0) = {ν | Im ν > T0, − 12 < Re ν < 1

2}. (6.1)

Since the Riemann hypothesis has already been rigorously established up to Im ν =O(106), we can simplify the calculations of this paper tremendously by taking T0

to be large. Initially, we take T0∼= 103.

The following lemma will be quite useful:

LEMMA 6.1. For any ν ∈ S(T0), and k such that Im k > −1/4, we have

|M(−)(ν, k) − M(−)0 (k)| � C2

T 20

(6.2)

and

C2 � 103. (6.3)


Proof. Taking the expansion of M(−)(ν; k) in powers of g = (ν2 − 1/4)−1 tofirst order, we have

M(−)(ν, k) − M(−)

0 (k) = gM(−)

1 (k) + gR(−)

1 (ν, k), (6.4)

where M(−)1 (k) is given by Equation (5.36), and

R(−)1 (ν; k) =

∫ ∞

1dα

(5∑

5=1

6(2)5 (α)

[α − ik]5)

[αν/2 + α−ν/2]. (6.5)

Here 6(2)5 is given by Equation (5.26) and Table I.

First, we have for Im k > −1/4,

|M(−)1 (k)| �

4∑5=1

|b5||1 − ik|5 �

[4∑

5=1

|b5| · ( 43)

5

]. (6.6)

Using Table II, we get

|M(−)

1 (k)| < 68, Im k > − 14 . (6.7)

The upper bound on R(−)1 for Im k > −1/4 is

|R(−)

1 (ν, k)| � 2∫ ∞

1α1/4

(5∑

5=1

|6(2)5 (α)|α−5( 4

3)5

), (6.8)

where we have used∣∣∣∣ α

α − ik

∣∣∣∣ < 43 for α � 1 and Im k > − 1

4 .

Using Equation (5.26), we have

|R(−)1 (ν, k)| � 2

∫ ∞

1dα

{5∑

5=1

( 43 )

5|5−5∑j=0

|C(2)(5, j)|α−1/2+jψ(j)(α)|}, (6.9)

where we note that C(n)(5, j) = (−1)5+1 · |C(n)(5, j)| as can be seen from Table I.The series in Equation (3.7) that defines ψ(α) is highly convergent for α � 1.

Indeed the first term gives a good approximation to it and to its first six derivatives.One can easily derive the bounds, 0 � j � 7,

πje−πα � |ψ(j)(α)| � πje−πα(1 + ε(j)), (6.10)

where ε(j) is

ε(j) = e−3π [1 + (2)2j ]. (6.11)

22 N. N. KHURI

For j � 4, ε(j) < 0.021. Thus it is sufficient for the purposes of this estimateto use ψ(j)(α) ∼= (−1)jπje−πα . Substituting this in (6.9) and carrying out the α

integration, we get

|R(−)1 (ν, k)|

� 2(1.1)√π

5∑5=1

( 43)

5|5−5∑j=0

|C(2)(5, j)|(−1)j+(j + 12 ;π)| ≡ C ′

2, (6.12)

where +(j, β) is the incomplete gamma function. From Table I, it is now easy tocheck our bound of C2 ≡ C ′

2+68 < 200. This completes the proof of Lemma 6.1. ✷It should be apparent to the reader that one could use more refined methods

to obtain a much better bound on R(−)

1 . We do not do this at this stage. Our mostimportant task is to study the Riemann conjecture for Im ν > T0 with T0 takenbelow the maximum for which the hypothesis has been rigorously established. In afuture paper, we will try to find the lowest value of T0 for which our method works.

The function M(−)

0 (k) given by Equation (5.13) is a rational function of k

M(−)

0 (k) = 1 + a1

1 − ik+ −1 − a1

(1 − ik)2

= −k[k + i(2 + a1)](1 − ik)2

. (6.13)

Here

a1 = −1 + 8ψ(1) = −0.6543. (6.14)

Obviously, M(−)

0 has two zeros,

k1 = 0 and k2 = −i(2 + a1) = −i(1 + 8ψ(1)).

Thus, Im k2 < −1. Hence, M(−)

0 (k) has only one zero in the half-plane, Im k � 0.Focusing on the domain Im k > −1/4, and |k| > 1/4 we get, with k = κ + iτ ,

|M(−)0 (k)| =

√κ2 + τ 2

√κ2 + (τ + η)2

κ2 + (τ + 1)2, (6.15)

where η = 2 + a1 > 1.It is easy to find a lower bound for |M(−)

0 | in the above domain. Setting q =(0,−η), and p = (0,−1) we get

|M(−)

0 (k)| = |k| · |k − q||k − p|2 � |k|2

|k − p|2 � 1

25. (6.16)

We can now prove the following lemma:


LEMMA 6.2. For any ν ∈ S(T0), and k such that Im k � 0, with |k| � 1/4,M(−)(ν, k) has no zeros and has a lower bound

|M(−)(ν, k)| >(

0.04 − 1

T0

). (6.17)

Proof. From Lemma 6.1 we have

|M(−)(ν, k)| > |M(−)0 (k)| − C2

T 20

, (6.18)

with C2/T0 < 2. Using (6.16) we get

|M(−)(ν, k)| >(

0.04 − 1

T0

)>

1

26. (6.19)

Thus any zeros, k0 of M(−)(ν, k) in the upper half k-plane must have |k0| < 1/4. ✷Proceeding further, we have the following lemma:

LEMMA 6.3. For any fixed ν ∈ S(T0), the maximum number of zeros of M(−)(ν; k)with Im k � 0 is one.

Proof. From Lemma 6.1 we get, for ν ∈ S(T0),

|M(−)(ν, k) − M(−)0 (k)| < 1

T0, |k| = 1

4 . (6.20)

By very similar arguments, we can also show that∣∣∣∣dM(−)(ν, k)

dk− dM(−)

0

dk

∣∣∣∣ < λ

T0, |k| = 1

4 , (6.21)

where λ = O(1).Let N(0)

14

denote the number of zeros of M(−)

0 (k) in the disc |k| � 1/4, and N 14

be the corresponding number for M(−)(ν, k), ν ∈ S(T0), then

N 14− N

(0)14

= 1

2πi

∮C 1

4

dk

{ [M(−)(ν, k)]′M(−)(ν, k)

− [M(−)0 (k)]′

M−0 (k)

}, (6.22)

where C 14

is the circle |k| = 1/4, and the prime indicates differentiation withrespect to k.

Hence,

|N 14− N

(0)14

|

� (26)2

4{Max|k|= 1

4|M(−)

0 (k)[M(−)(ν, k)]′ − [M(−)0 (k)]′M(−)(ν, k)|}

�(

169λ

T0

)< 1, (6.23)

24 N. N. KHURI

where we have used Equations (6.16), (6.19), and (6.21). Since M(−)0 (k) has only

one zero in the disc, so does M(−)(ν, k) and Lemma 6.2, which proves the absenceof zeros with Im k � 0, and |k| � 1/4, completes our proof. ✷

The next question is where is this one zero of M(−)(ν; k)? To answer thisquestion we first-order expansion for M(−)(ν, k),

M(−)(ν, k) = M(−)

0 (k) + gM(−)

1 (k) +

+ g

∫ ∞

1dα

(5∑

5=1

6(2)5 (α)

(α − ik)5

)[αν/2 + α−ν/2]. (6.24)

With k = 0, we have

2ξ(ν + 1/2) = g

∫ ∞

1dα

(5∑

5=1

6(2)5 (α)

α5

)· [αν/2 + α−ν/2]. (6.25)

Subtracting these two equations, we get

M(−)(ν, k)

= 2ξ(ν + 1/2) + M(−)

0 (k) + gM(−)

1 (k)+

+ g

∫ ∞

1dα

(5∑

5=1

{1

(α − ik)5− 1

α5

}6

(2)5 (α)[αν/2 + α−ν/2]

). (6.26)

Note that now the integral on the right-hand side is also O(k) as k → 0.The ξ function for large values of | Im ν| is exponentially small, mainly due to

the +(s/2) factor in Equation (3.4). In fact given standard results on the order ofζ(s) in the critical strip we have

ξ( 12 + ν) = O(|ν|pe

−| Im ν|π4 ),

where p = 2 + δ and 0 < δ < 1. Thus, from Equation (6.26) and the exactexpression for M(−)

0 (k) in Equation (6.13), we get the position of the zero, k0, nearthe origin,

k0∼= −2iξ( 1

2 + ν)

[(2 + a1) + O( 1T 2

0)] + O(k2

0), (6.27)

where (2 + a1) > 1. Hence,

Im k0∼= −2 Re ξ( 1

2 + ν)

2 + a1. (6.28)

If Re ξ( 12 + ν) � 0, M(−)(ν, k) has no zeros for Im k > 0. On the other hand, if

ν ∈ S(T0) is such that Re ξ(1/2 + ν) < 0 there will be one zero close to the origin,but in the upper half k-plane.


Finally, we note one important fact, namely, if (ν0 + 1/2) is a zero of the ξ -function, then M(−)(ν0, k) has no zeroes for Im k > 0, and its only zero withIm k > −ε occurs exactly at k = 0.

In summary M(−)(ν; k) , with ν ∈ S(T0), has most of the properties of a Martintype Jost function with the exception of one, i.e. reality. We list these properties:

(i) M(−)(ν, k) analytic in the cut k-plane with a cut for k = iτ , −∞ < τ < −1.(ii) lim|k|→∞ M(−)(ν, k) = 1.

(iii) M(−)(ν, k) has no zeros for Im k > 1/4, and ν ∈ S(T0�).(iv) If we write

>(ν, u) ≡ (ν2 − 14 )

∫ ∞

1dαψ(α)α1/4[αν/2 + α−ν/2]e−αu, (6.29)

then

M(±)(ν, k) = 1 +∫ ∞

0>(ν, u)e∓iku du. (6.30)

In fact we could have used Equations (6.29) and (6.30) as the starting defini-tions of M(±)(ν, k).

(v) We define the S-matrix

S(ν, k) = M(+)(ν, k)

M(−)(ν, k), (6.31)

and it follows that∫ +∞

−∞|S(ν, k) − 1|2 dk < ∞. (6.32)

(vi) The reality condition does not hold for all ν. If ν is purely imaginary, i.e.ν = it , then for real k, (M(+)(ν, k))∗ = M(−)(ν, k) and, hence, |S(ν, k)| = 1.However, if ν is complex, i.e. Re ν �= 0, then the above relation does not hold.However, we still have for real k

|S(ν, k)| = 1 + O

(1

|ν|2). (6.33)

For an arbitrary ν, ν ∈ S(T0), we can still carry out the inverse scatteringprogram by properly handling the one zero in the upper half k plane. We will dothat in Section 8, where the resulting potential is complex.

7. The Limit Case, |ν| → ∞Before we proceed to the main proof, we shall solve exactly the limiting case |ν| →∞. This result will be extremely useful in the rest of this paper.

26 N. N. KHURI

We start with

M(±)(ν, k) → M(±)

0 (k), |ν| → ∞, (7.1)

where, from Equation (6.13), we have

M(±)0 (k) = −k[k ∓ i(2 + a1)]

(1 ± ik)2. (7.2)

We thus have Jost functions which are rational in k. This is the case first studiedby Bargmann [22] in the paper which gave the famous phase equivalent potentials.

The S-matrix is also rational,

S0(k) ≡ M(+)

0

M(−)

0

=[k − i(2 + a1)

k + i(2 + a1)

](1 − ik

1 + ik

)2

, (7.3)

where (2+a1) > 1, a1 = −1+8ψ(1). One can use Bargmann’s method to uniquelydetermine a potential V0(x) which has the S-matrix given here. But we prefer todetermine V0 by using Marchenko’s method.

The Marchenko kernel F0 is

F0(x) = 1

2π

∫ ∞

−∞(S0(k) − 1)eikx dk. (7.4)

This Fourier transform converges in the mean, (S0 − 1) → O(1/k) as k → ±∞.By contour integration

F0(x) = λ0e−x + λ1xe−x , (7.5)

where

λ0 = 8a1 + 4a21 − 4

(3 + a1)2(7.6)

and

λ1 = −4(1 + a1)

(3 + a1). (7.7)

Both λ0 and λ1 are negative.The Marchenko equation is

A0(x, y) = F0(x + y) +∫ ∞

x

A0(x, u)F0(u + y) du (7.8)

and A0(x, y) ≡ 0 for y < x. With F0 as defined by Equation (7.5), one can easilyobtain the exact solution of the integral equation (7.8).

From Bargmann’s paper [22], it is clear that we have the ansatz

A0(x, y) ≡ [B(x) + (y − x)C(x)]e−(y−x). (7.9)


Substituting this trial solution in Equation (7.8) and carrying out the integrationover u, we get

A0(x, y) = e−(y−x)

{e−2x

[(B(x)

2+ C(x)

4

)(λ0 + λ1x) +

+ B(x) + C(x)

4λ1 + λ0 + xλ1

]+

+ ye−2x

[B(x)λ1

2+ C(x)λ1

4+ λ1

]}. (7.10)

But from Equation (7.9) we also have

A0(x, y) = [(B − xC) + yC]e−(y−x). (7.11)

Comparing the coefficients of y in Equations (7.10) and (7.11) we get

e2xC(x) = B(x)λ1

2+ C(x)λ1

4+ λ1, (7.12)

and the terms to zero-order in y give(B

2+ C

4

)(λ0 + λ1x) + Bλ1

4+ Cλ1

4+ λ0 + xλ1 = e2x[B − xC]. (7.13)

These last two equations determine B(x) and C(x) giving

C(x) = λ1 − (λ21/4)e−2x

[e2x − (λ2

116 )e

−2x − 12(λ0 + λ1) − λ1x]

(7.14)

and

B(x) = λ0 + 2λ1x + (λ2

14 )e

−2x

[e2x − (λ2

116)e

−2x − 12 (λ0 + λ1) − λ1x]

. (7.15)

Since λ1 < 0, λ0 < 0, and (λ21/16) � 1, the denominators in Equations (7.14)

and (7.15) do not vanish for any x � 0.One can simplify Equations (7.14) and (7.15) by defining

ρ ≡ −λ1

4=(

1 + a1

3 + a1

)= 0.1474. (7.16)

Then its is easy to show that

− 12 (λ0 + λ1) = 1 − ρ2. (7.17)

We obtain

C(x) = −(4ρ + 4ρ2e−2x)

[e2x − ρ2e−2x + (1 − ρ2) + 4ρx] (7.18)

28 N. N. KHURI

and

B(x) = (2ρ2 + 4ρ − 2) − 8ρx + 4ρ2e−2x

[e2x − ρ2e−2x + (1 − ρ2) + 4ρx] . (7.19)

From Equation (7.9) we see that the potential, V0(x), is

V0(x) = −2dA0

dx(x, x) = −2

dB(x)

dx. (7.20)

This leads to

(4 − 2ρ2 − 16ρ)e2x − 4(1 − ρ2)ρ2e−2x −V0(x) = − 16ρx(e2x + ρ2e−2x) − 64ρ2x − 16ρ2

[e2x − ρ2e−2x + (1 − ρ2) + 4ρx]2. (7.21)

Note that V0(x) = O(e−2x) as x → +∞.The two Jost solutions f (±)

0 are given by

f(±)0 (k, x) = e∓ikx +

∫ ∞

x

dyA0(x, y)e∓iky . (7.22)

Substituting Equation (7.9) for A0, we get

f ±0 = e∓ikx

{1 + B(x)

(1 ± ik)+ C(x)

(1 ± ik)2

}. (7.23)

One can now check directly that

−d2f ±0

dx2+ V0(x)f

±0 = k2f ±

0 , (7.24)

also f(±)0 → M±

0 (k) as x → 0.The results of this section can also be obtained by using the technique developed

by Bargmann [22] which preceded the results of [6] and [7]. The Jost functionsdefined in Equation (7.2) do indeed uniquely determine the potential V0(x) andits solutions f

(±)0 (k, x). The fact that M(±)

0 (k) = O(k), as k → 0, leads to thedegenerate case using Bargmann’s method, but the final results agree with those byMarchenko’s method.)

Finally, it should be remarked that the full scattering amplitude for V0(x) has apole at k = 0. However, this is not part of the point spectrum. There is no L2(0,∞)

solution of the Schrödinger equation with V0(x) for k = 0.

) We thank H. C. Ren for clarifying and checking this point.


8. The Marchenko Equation

We define an S-matrix as in Equation (6.34)

S(ν, k) = M(+)(ν, k)

M(−)(ν, k), (8.1)

where ν ∈ S(T0). S(ν, k) is, for any ν, analytic in k in the strip −1 < Im k < +1.Next we define a Marchenko kernel for S(ν, k),

F(ν; x) = 1

2π

∫L

dk[S(ν, k) − 1]eikx , x > 0, (8.2)

where L is a line Im k = δ > 0, 1 > δ > 0, and without loss of generality we fixδ, δ = 1/4. This Fourier transform is convergent in the mean (S − 1) = O(1/k),as Re k → ±∞.

Actually, we can also perform an integration by parts on (8.2) for any x > ε >

0. Since dS/dk is bounded and (dS/dk) = O(1/k2) as k → ±∞, we have absoluteconvergence for any x > 0.

It is important to note here that as shown in Equation (6.32), [S(ν, k) − 1] ∈L2(−∞,+∞), along the line Im k = 1/4.

Of course, Equation (8.2) is not the standard definition of the Marchenko kernel.In the standard case, one integrates along the real k-axis. If we move the contourin (8.2) to the real axis, then there could be an extra contribution from the poleproduced by the zero of M(−)(ν, k) when Re ξ(ν+1/2) < 0. But for a Martin typeS-matrix, all the scattering data, including that coming from the point spectrum, iscontained in the discontinuity across the branch cut on the imaginary k-axis (see[19–21]).

We prove the following lemma:

LEMMA 8.1. F(ν, x) is(a) continuous and differentiable in x, x ∈ [0,∞);(b) F(ν, x) = O(e−x) as x → +∞;(c) Both F(ν, 0) and F ′(ν, 0) are finite and∫ ∞

0|F(ν, x)| dx < ∞,

∫ ∞

0|F ′(ν, x)| dx < ∞; (8.3)

(d) F(ν, x) is analytic in ν, for ν ∈ S(T0), and fixed x � 0.Proof. In the Appendix, we prove that F(ν, x) can be written as

F(ν, x) = 1

π

∫ ∞

1D(ν, α)e−αx dα, (8.4)

where

D(ν, α) = π(ν2 − 1/4)ψ(α)α1/4[αν/2 + α−ν/2]M(−)(ν, iα)

. (8.5)

30 N. N. KHURI

This result is obtained by deforming the contour in Equation (8.2) and using theoriginal representation (3.10) for M(±)(ν, k). We note that S(ν, k) is analytic forIm k > 0, except on the cut k = iτ ; 1 � τ < ∞. Next we note that

|M(−)(ν; iα)| �∣∣∣∣α(α + (2 + a1))

(1 + α)2

∣∣∣∣− |g||M(−)1 (iα) + R

(−)1 (ν; iα)|. (8.6)

This follows from Equations (6.4) and (6.13). Since (2 + a1) > 1, we get

|M(−)(ν, iα)| � 12 + 1

2

[2 + a1

(1 + α)

]− C2

T 20

, α � 1, (8.7)

where the last term comes from Lemma 6.1. Hence, we have

|M(−)(ν, iα)| � 12 . (8.8)

Finally, we obtain from (8.5), with |Re ν| < 1/2,

|D(ν, α)| � 2π(ν2 − 14 )e

−παα1/2, α � 1. (8.9)

This bound guarantees the absolute and uniform convergence of the Laplace trans-form in Equation (8.4) for all x ∈ [0,∞) and, hence, all the assertions (a), (b),and (c) of our lemma are true. Finally, (d) is also true, given the lower bound inEquation (8.8) and the uniform bound on (αν/2 + α−ν/2) for ν ∈ S(T0). ✷

The Marchenko equation can now be defined as

A(ν; x, y) = F(ν; x + y) +∫ ∞

x

duA(ν; x, u)F (ν;u + y), (8.10)

with

A(ν; x, y) ≡ 0, y < x. (8.11)

The integral equation (8.10) is of Fredholm type and the Hilbert–Schmidt normof F is finite,

F 2 =∫ ∞

x

du∫ ∞

x

dv|F(ν;u + v)|2 < ∞. (8.12)

This follows from Lemma 8.1.We first prove the following lemma:

LEMMA 8.2. For all ν ∈ S(T0) and x � 0, we have

|F(ν; x) − F0(x)| < C

|Im ν|2 e−x/4, (8.13)

where the constant C is bounded

C < 2 × 103. (8.14)


Proof. From Equations (6.4) and (6.5), we have

M(±)(ν, k) − M(±)0 (k) = gM

(±)1 (k) + gR

(±)1 (ν; k), (8.15)

where g = (ν2 − 1/4)−1. From the definitions of F(ν, x) and F0(x), we get

F(ν, x) − F0(x) = 1

2π

∫L

dk

[M(+)(ν, k)

M(−)(ν, k)− M

(+)0 (k)

M(−)0 (k)

]eikx . (8.16)

This gives

F − F0 = (EF)1 + (EF)2, (8.17)

with

(EF)1 ≡ g

2π

∫L

eikx(M

(−)0 M

(+)1 − M

(+)0 M

(−)1 )

M(−)0 M(−)

(8.18)

and

(EF)2 ≡ g

2π

∫L

eikx(M

(−)

0 R(+)

1 − M(+)

0 R(−)

1 )

M(−)

0 M(−). (8.19)

Again, here L is the line k = (λ + (i/4)), and −∞ < λ < +∞.In order to get bounds on (EF)1,2, we need to separate out the terms which

are only conditionally convergent, i.e. O(1/k) as k → ∞, from those which areabsolutely convergent and, hence, easier to handle.

From Equations (5.36) and (6.5), we write

M(±)1 (k) = b1

1 ± ik+ M

(±)1 (k) (8.20)

and

R(±)

1 (ν, k) =∫ ∞

1dα

6(2)1 (α)

α ± ik(αν/2 + α−ν/2) + R

(±)

1 (ν, k), (8.21)

where

M(±)

1 (k) =4∑

5=2

b5

(1 ± ik)5, (8.22)

R(±)1 (ν, k) =

∫ ∞

1dα

5∑5=2

6(2)5 (α)

(α ± ik)5[αν/2 + α−ν/2]. (8.23)

Both M1 and R1 are O(1/k2) as |k| → ∞.

32 N. N. KHURI

For (EF)1, we can write

(EF)1 ≡ (EF)11 + (EF)12, (8.24)

with

(EF)11 ≡ gb1

2π

∫L

dkeikx[M

(−)

0 (k)

(1 + ik)− M

(+)

0 (k)

(1 − ik)

](1

M(−)

0 M(−)

)(8.25)

and

(EF)12 = g

2π

∫L

dkeikx[M

(−)

0 (k)M(+)

1 (k) − M(+)

0 (k)M(−)

1 (k)

M(−)

0 (k)M(−)(ν, k)

]. (8.26)

The integral in (8.25) is conditionally convergent, |M±0 (k)| → 1 as |k| →

∞, and |M(−)(ν, k)| is bounded from below for all k with Im k � 1/4. Also|M(−)(ν, k)| → 1 as |k| → ∞.

To obtain a bound on (EF)11 we first note the following:∫L

dkeikx{(

M(−)

0 (k)

1 − ik

)1

M(−)

0 (k)M(−)(ν, k)

}= 0. (8.27)

This follows from Jordan’s lemma. The integrand in (8.27) is analytic for Im k � 0,and the bracketed term is O(1/k) for large |k|.

Adding twice the left-hand side of Equation (8.27) to Equation (8.25), oneobtains

(EF)11 = gb1

2π

∫L

dkeikx[

2M(−)0 (k)

1 + k2− M

(+)0 (k) − M

(−)0 (k)

1 − ik

]×

×(

1

M(−)

0 (k)M(−)(ν, k)

). (8.28)

To obtain an upper bound on |(EF)11|, we first need lower bounds on M(−)0 (k)

for k ∈ L.From Equation (5.13), we have

|M(−)

0 (k)| � 1 − |a1||1 − ik| − |a2|

|1 − ik|2 , (8.29)

with

|a1| = 0.654, |a2| = 0.346 and k = λ + i

4, −∞ < λ < +∞,

we finally have

|M(−)0 (k)| � 0.255 > 1

4 , k ∈ L. (8.30)


Lemma 6.1 will now give us a lower bound for M(−)(ν, k), ν ∈ S and k =λ + i/4. Since |M(−) − M

(−)

0 | � C2/T2

0 we get

|M(−)(ν, k)| � 14 , (8.31)

for k = λ + i/4.Finally, we need to bound M

(+)

0 − M(−)

0 , where from Equation (5.13),

M(+)0 − M

(−)0 = −2ika1

1 + k2− 4ika2

(1 + k2)2. (8.32)

But for k = λ + i/4, |k/(1 + ik)| < 1, we obtain

|M(+)0 (k) − M

(−)0 (k)| � δ

|1 − ik| , (8.33)

with

δ = 2|a1| + 6415 |a2| ∼= 2.78. (8.34)

The above bounds lead us immediately to

|EF |11 � 2|g||b1|e−x/4

π

∫ +∞

−∞dλ

{2

|1 + (λ + i/4)2| + 4δ

|1 − ik|2}

(8.35)

� c11

| Im ν|2 e−x/4, (8.36)

where

c11 = 4|b1|π

(2 + 4δ) ·∫ ∞

0

dλ1615 + λ2

� 290. (8.37)

The bound for (EF)12 is easier to calculate, from (8.26)

|(EF)12| � |g|e−x/4

2π

∫ +∞

−∞dλ

{4|M(+)

1 (k)| + 4|M(−)

1 (k)| ·∣∣∣∣M

(+)0

M(−)0

∣∣∣∣}. (8.38)

For k = λ + i/4, a simple calculation gives∣∣∣∣M(+)0 (k)

M(−)0 (k)

∣∣∣∣ � 5/3. (8.39)

Using Equation (8.22), we get

|(EF)12| � c12e−x/4

| Im ν|2 , (8.40)

34 N. N. KHURI

with

c12 = 4

π

(4∑

5=2

|b5|)∫ ∞

0dλ

[1

[( 34 )

2 + λ2] + ( 53 )

1

[( 54)

2 + λ2]]

∼= 112. (8.41)

The values of b5 are given in Equation (5.40).To estimate (EF)2 we again split R(±)(ν, k) into two terms as in Equation (8.21).

We write

(EF)2 ≡ (EF)21 + (EF)22, (8.42)

where

(EF)21 = g

2π

∫L

dkeikx[M

(−)

0 (k)r(+)

1 − M(+)

0 (k)r(−)

1

M(−)

0 (k)M(−)(ν, k)

](8.43)

and

(EF)22 = g

2π

∫L

dkeikx[M

(−)0 (k)R

(+)1 − M

(+)0 R

(−)1

M(−)0 (k)M(−)(ν, k)

], (8.44)

with R(±)1 given in Equation (8.23) and

r(±)1 (ν, k) =

∫ ∞

1dα

6(2)1 (α)

α ± ik[αν/2 + α−ν/2]. (8.45)

Equation (8.43) can be rewritten as

(EF)21 = g

2π

∫L

dkeikx{M

(−)0 (r

(+)1 + r

(−)1 )

M(−)0 M(−)

− (M(+)0 − M

(−)0 )r

(−)1

M(−)0 M(−)

}. (8.46)

Here we have again used Jordan’s lemma, which implies∫L

dkeikx[M

(−)0 r

(−)1

M(−)0 M(−)

]≡ 0. (8.47)

From Equations (8.45) and (8.46) we get

|(EF)21| � |g|e−x/4

2π

∫ +∞

−∞dλ∫ ∞

1dα(2α1/4)|6(2)

1 (α)| ×

×{

8α

|α2 + k2| + 16|M(+)

0 − M(−)

0 ||α + 1/4 − iλ|

}, (8.48)

where k = λ + i/4. Using the bound (8.33) we obtain

|(EF)21| � c21e−x/4

| Im ν|2 , (8.49)


with

c21 = 1

2π

∫ ∞

1dα(2α1/4)|6(2)

1 (α)| ×

×∫ +∞

−∞dλ

[8α

(α2 − 116) + λ2

+ 16δ

|α + 1/4 − iλ|| 54 − iλ|

]. (8.50)

This leads to

c21 �∫ ∞

1dαα1/4|6(2)

1 (α)|[ 12815 + 64

5 δ] � (45)∫ ∞

1dαα1/4|6(2)

1 (α)|. (8.51)

Using the definition of 6(2)1 (α) in Equation (5.26), Table I, and the bounds on

ψ(j)(α) given in Equation (6.10), one can easily get a rough numerical bound onthe above integral,

∫∞1 dαα1/4|6(2)(α)| � 10 and, hence,

c21 � 450. (8.52)

From Equations (8.44), (8.30), (8.31), and (8.23), we get

|(EF)22| � 4|g|4

e−x/45∑

5=2

∫ ∞

1dαα1/4|6(2)

5 (α)| ×

×∫ +∞

−∞dλ

{1

|α + ik|5 +∣∣∣∣M

(+)

0

M(−)

0

∣∣∣∣ 1

|α − ik|5}

(8.53)

with k = λ + i/4. Using the fact that

|α ± ik| � [(α ∓ 14)

2 + λ2]1/2 and

∣∣∣∣M(+)0

M(−)0

∣∣∣∣ < 53 for k ∈ L,

we have

|(EF)21| � c22

|Im ν|2 e−x/4 (8.54)

with

c22 � 32

3π

5∑5=2

∫ ∞

1dαα1/4|6(2)

5 (α)|(

1

α − 1/4

)5−1

β5, (8.55)

where

β5 ≡∫ +∞

−∞du

(1 + u2)5/2. (8.56)

For α � 1, |(α/(α − 1/4)| < 4/3 and, hence,

c22 � 32

3π

5∑5=2

A5β5(43)

5, (8.57)

36 N. N. KHURI

with

A5 =∫ ∞

1dαα5/4−5|6(2)

5 (α)|. (8.58)

A simple numerical estimate will give

c22 � 103. (8.59)

This completes the proof of our lemma with C = c11 +c12 +c21 +c22 < 2×103. ✷Lemma 8.2 guarantees that for ν ∈ S, F = F0 + O(1/T 2

0 ). Indeed, for theHilbert–Schmidt norms, we have

F � F0 + F − F0 . (8.60)

However, from the lemma,

F − F02 =

∫ ∞

x

du∫ ∞

x

dv|F(ν, u + v) − F0(u + v)|2

� C2

|ν|4∫ ∞

x

du∫ ∞

x

dve−(u+v)/2

� C2

|ν|4∫ ∞

0we−w/2dw. (8.61)

Thus

F − F0 � 2C

T 20

� 2

T0. (8.62)

We have the exact expression for F0(x) given in Equation (7.5) and we cancalculate F0 exactly. The Hilbert–Schmidt norm for F0 is

F02x =

∫ ∞

x

du∫ ∞

x

dv[F0(u + v)]2. (8.63)

Note that here x appears as a parameter (see Equation (8.10)). As x increases thenorm of F0 tends to zero. From Equation (7.5), we get

F02x=0 =

∫ ∞

0dw[λ2

0w + 2λ0λ1w2 + λ1w

3]e−2w, (8.64)

with λ0 and λ1 given by Equations (7.6) and (7.7). After some algebra, we obtain

F02x=0 = 1 + λ2

1

16, (8.65)

with λ1 = −0.590. Thus the Hilbert–Schmidt norm for x = 0 is slightly biggerthan one, however, it is easy to show that for x > x0

∼= 0.1,

F0 < 1. (8.66)


Hence, the iterative series for both F and F0 converges for x > 0.1. But toprove the existence of A(ν; x, y) for all x � 0, we will use Equation (8.61) andproceed in another way.

In Section 7, we gave an explicit solution of the integral equation

A0 = F0 + A0F0. (8.67)

We can now write

I0 ≡ (1 − F0)−1, (8.68)

and

A0 = (1 − F0)−1F0 = I0F0. (8.69)

This leads to

A0 + 1 = I0. (8.70)

The kernel A0(x, y) is given explicitly in Equation (7.9) and (7.18)–(7.19), andthus I0(x, y) is known.

The full Marchenko equation (8.10) can now be written as

A = F + AF. (8.71)

We define I as

I = (1 − F)−1 (8.72)

and

A = (1 − F)−1F. (8.73)

We prove that both I and A exist and have a finite norm. We have

A + 1 = I = (1 − F)−1. (8.74)

To show that I (and A) exist, we note first that

I = (1 − F)−1 = (1 − F0 − E)−1, (8.75)

where

E ≡ F − F0. (8.76)

Using the fact that (1 − F0)I0 = 1, we get

I = [(1 − F0)I0{(1 − F0) − E}]−1

= (1 − I0E)−1I0. (8.77)

38 N. N. KHURI

Using Equation (8.74), we get

A = (1 − I0E)−1I0 − 1

= (1 − I0E)−1(A0 + I0E), (8.78)

where A0 + 1 = I0.Next we define the operator K as

K ≡ I0E = (A0 + 1)E (8.79)

and obtain

A = A0 + (1 − K)−1(1 + A0)K. (8.80)

The Hilbert–Schmidt norm of E is small. Indeed, using Equation (8.61), wehave

E 2 ≡ F − F02 � C2

|ν|4 � 2

T 20

� 1. (8.81)

However, K ≡ A0E + E, and we get

K � A0 E + E . (8.82)

A0 is known and A02 < 5, thus

K � 3C

|ν|2 � 1, (8.83)

for all ν ∈ S(T0). Thus the inverse (1 −K)−1 is given by an absolutely convergentseries

(1 − K)−1 =∞∑n=0

Kn (8.84)

and has a bounded norm, (1 − K)−1 � 2.The final result for A is

A = A0 + H + A0H, (8.85)

where

H ≡∞∑n=1

Kn, (8.86)

and H � 2 K � 1. Indeed, we have A − A0 � C/|ν|2.The kernel K(ν; x + y) can be written as

K(ν; x, y) = F(ν; x, y) − F0(x + y) ++∫ ∞

x

duA0(x, u)[F(ν;u + y) − F0(u + y)]. (8.87)

The properties of the kernel K(ν; x, y) are similar to those of F(ν; x, y). Wehave the following lemma:


LEMMA 8.3. K(ν; x, y) is for y � x � 0, (a) analytic for ν ∈ S(T0);

(b) differentiable in both x and y;(c) analytic for Re x � 0 and Re y � 0, when ν ∈ S(T0); and(d) |K(ν; x, y)| � C/|ν|2e−(

x+y4 ).

Proof. These results follow from Equation (8.88), Lemma 8.1, and the exactresult (7.11) for A0(x, u). We note that the denominators appearing in the expres-sions (7.18) and (7.19) for B(x) and C(x) do not vanish for Re x � 0. Both B andC are thus analytic in the half plane Re x � 0.

The full expression for A(ν; x, y) is

A(ν; x, y) = A0(x, y) + H(ν; x, y) +∫ ∞

x

A0(x, u)H(ν;u, y), (8.88)

where

H(ν; x, y) =∞∑n=1

K(n)(ν; x, y) (8.89)

and

K(n)(ν; x, y)=∫ ∞

x

du1 . . .

∫ ∞

x

dun−1K(ν; x, u1)K(ν;u1, u2) . . . K(ν;un−1, y). (8.90)

The series in (8.89) is absolutely and uniformly convergent for y � x � 0, and allν ∈ S(T0). ✷

The properties of A(ν; x, y) can be summarized in the following lemma:

LEMMA 8.4. (a) For y � x � 0, A(ν; x, y) is analytic in ν for ν ∈ S(T0);(b) A(ν; x, y) is differentiable in both x and y, y � x. Also A(ν, 0, 0) and

[d/dx(A(ν; x, x)]x=0 are finite;(c) For fixed ν, ν ∈ S(T0), A(ν; x, y) is analytic in x and y for Re x � 0,

Re y � Re x � 0; and(d) For all ν ∈ S(T0), we have the bound

|A(ν; x, y) − A0(x, y)| � C

|ν|2 e−(x+y

4 ). (8.91)

Proof. These results follow immediately from Equation (8.89) and Lemmas 8.1and 8.3. The bound (8.91) follows from the bound (8.13) of Lemma 8.2. Theconstant C is certainly such that, C < 104, which is sufficient for our purposesat this stage, but can be improved with more careful estimates. ✷

40 N. N. KHURI

The next step is to define two functions U(ν; x) and f (±)(ν; k, x) as follows:

U(ν; x) = −2d

dxA(ν : x, x), x � 0 (8.92)

and

f ±(ν; k, x) = e∓ikx +∫ ∞

x

dyA(ν; x, y)e∓iky . (8.93)

Without recourse to the standard methods of inverse scattering, one can directlyprove the next lemma.

LEMMA 8.5. For any ν ∈ S(T0), f ± satisfy a Schrödinger equation with U(ν, x)

as the potential

−d2f ±

dx2+ U(ν; x)f ± = k2f ±. (8.94)

Proof. From (a) and (b) in Lemma 8.4 it follows that U(ν; x) is analytic in ν forν ∈ S(T0) and x � 0. Similarly, from Equation (8.94), it follows that f (−)(ν; k, x)(with x � 0) and Im k � 0 is also analytic in ν in the truncated strip. Similarly,f (+) with Im k � 0 is analytic. The same is true for d/dx(f (±)), and d2/dx2(f (±))

since absolute and uniform convergence allows us to differentiate under the integralsign in (8.93). ✷

In the next section we will prove the validity of Equation (8.94) on the lineν = it , t � T0. Hence, by analytic continuation, the Schrödinger equation (8.94)holds for all ν ∈ S(T0).

In the Appendix we will give a more direct proof of Equation (8.94) and alsoshow that one does indeed recover the original Jost function from the potentialU(ν; x).

9. The Case ν = it

For purely imaginary ν, our S-matrix, S(ν, k), is unitary and satisfies all the prop-erties needed for the standard inverse scattering methods of Gelfand, Levitan, andMarchenko to be applicable. We sketch some relevant results in this section.

First, we define S as

S(it, k) ≡ M(+)(it, k)

M(−)(it, k), t > π2. (9.1)

For real k, it follows from Equation (3.10) that

[M(+)(it, k)]∗ = M(+)(it,−k) = M(−)(it, k). (9.2)


S(it, k) satisfies all the conditions given in Faddeev’s [14] review paper, whichare sufficient to guarantee that the Marchenko equation will lead to a unique realpotential U(it, x). We can easily check that for real k,

|S(it, k)| = S(it, 0) = S(it,∞) = 1 (9.3)

and

[S(it, k)]∗ = S(it,−k). (9.4)

The number of discrete eigenvalues of S for fixed t > π2 is at most one (seeSection 6). In the physicist’s language, we have either one bound state or oneantibound state. This is evident from Equation (6.28) and the fact that ξ(it + 1/2)is real. We will discuss this point in more detail at the end of this section.

The Marchenko kernel is now given by

F(it, x) = 1

2π

∫ ∞

−∞dk[S(it, k) − 1]eikx; x > 0 (9.5)

for the case

ξ(it + 12 ) � 0 (9.6)

and

F(it, x) = 1

π

∫ +∞

−∞dk[S(it, k) − 1]eikx + c0e−τ0x (9.7)

for the case where ξ(it + 1/2) < 0, i.e. with a bound state at E = −τ 20 . Both

Fourier transforms in Equations (9.5) and (9.6) are convergent in the mean, since[S − 1] = O(1/k) for large k. Also, it is clear that F(it, x) is real.

As noted previously ξ(it + 1/2) = O(tpe− πt4 ) and, hence, small for, t > π2,

this makes τ0 � 1/4 and the bound state is very shallow for t > π2. One can nowmove the contour of integration up in both Equations (9.5) and (9.6) to obtain

F(it, x) = 1

2π

∫L

dk[S(it, k) − 1]eikx, x > 0 (9.8)

for both cases. Here L is the line Im k = 1/4. In the case of Equation (9.6), thecontribution from the pole at k = iτ0 exactly cancels the second term on the right-hand side, see [22].

The solution of the Marchenko equation, A(it; x, y) exists, is real, and differ-entiable for y � x > 0. The resulting potential, U(it, x), is real, continuous for allx � 0, and O(e−2x) for large x.

We close this section by calculating the position of the bound state or antiboundstate for fixed ν = it , t > T0.

42 N. N. KHURI

Rewriting Equation (6.26) for ν = it , we get

M(−)(it, k)

= 2ξ(it + 12 ) + M

(−)0 (k) + gM

(−)1 (k)+

+ 2g∫ ∞

1dα

(5∑

5=1

{1

(α − ik)5− 1

α5

}6

(2)5 (α)

[cos

(t

2logα

)]), (9.9)

where

g = −1

t2 + 1/4. (9.10)

Now, ξ(it + 1/2) is real and exponentially small for large t . The three other termson the right-hand side of Equation (9.8) are all O(k) for small k.

From Equation (6.13), we have

M(−)0 (k) = −ik(2 + a1) + O(k2), (9.11)

with

a1 = −1 + 8ψ(1) = −0.6543. (9.12)

Thus, for t > T0, the one zero of M(−)(it, k) will occur only when k = iτ and

τ = − 2ξ(it + 1/2)

(2 + a1) + O( 1t2 )

+ O(τ 2), (9.13)

where the O(1/t2) term is real, and the same for the O(τ 2) term. Note that M(−)0 (iτ )

and M(−)1 (iτ ) are both real as is the integral in (9.8) for k = iτ .

The resulting potential, or one parameter family of potentials,

U(it, x) ≡ V (g, x), (9.14)

has a remarkable property as t increases, t > T0. It will have exactly one boundstate when ξ(it + 1/2) < 0, with energy E0 = −τ 2

0 ,

E0 = − 4[ξ(it + 1/2)]2

[(2 + a1) + O( 1t2 )]2

+ O([ξ(it + 1/2)]3). (9.15)

Then, as we pass a Riemann zero and ξ(it +1/2) > 0, there will be no bound stateuntil t reaches the next Riemann zero.

As t → +∞, the potential, U(it, x), presents us with a seemingly puzzlingsituation. The bound state, i.e. a point spectrum of one, appears and then as t

increases disappears, with this process repeating as t increases until as t → ∞,we reach V0(x) which has no point spectrum.

Schwinger’s theorem relating the number of bound states to the number ofnodes of the zero energy regular solution, φ(it; 0; x), defined in Equation (2.8),is instructive for the present case.


From [13], we have an integral equation for φ,

φ(it; k; x) = sin kx

k+∫ x

0

sin k(x − x′)k

V (x′)φ(it; k; x′) dx′, (9.16)

where clearly φ(it; k; 0) ≡ 0. The zero energy φ is given by

φ(it; 0; x) =[

1 +∫ x

0V (x′)φ(it; 0; x′) dx′

]x −

−∫ x

0V (x′)φ(it; 0, x′) dx′. (9.17)

For x not large, x � T0, we can approximate φ by the t → ∞ solution

φ(it; k; x) = φ0(k, x) + O

(1

t2

), (9.18)

where φ0 is defined as in (2.8) but with f ± replaced by f ±0 and M± replaced by

M±0 . It is easy to check that φ0(0, x) is positive for x not large, and φ0(0; 0) = 0.

But from Equation (9.17) we see that the large x behavior is

φ(it; 0; x) → [C(t) + o(1)] ++ x

[1 +

∫ ∞

0V (x′)φ(it; 0; x′) dx′ + o(1)

], (9.19)

where

C(t) = −∫ ∞

0x′V (x′)φ(it; 0; x′) dx′. (9.20)

But under the integral sign, we can replace V by V0 and φ by φ0, and obtain

C(t) = −∫ ∞

0x′V0(x

′)φ0(it; 0, x′) dx′ + O

(1

t2

), (9.21)

where V0 and φ0 are known exactly from Section 7. It is a simple matter to checkthat

C(t) > 0, t > T0. (9.22)

Next in [13], we have the result relating the Jost function to φ, and it gives

M(−)(it; k) = 1 +∫ ∞

0dx′V (it; x′)φ(it; k; x′). (9.23)

Taking the k → 0 limit, we have from Equation (9.19)

φ(it; 0; x) → C(t) + 2ξ(it + 12 )x, x → ∞. (9.24)

44 N. N. KHURI

We can only have a node in φ if ξ(it + 1/2) < 0, otherwise there is no node andno bound state. For t > T0 and large, the node occurs at large values of x,

x0∼= C(t)

2ξ(it + 12 )

∼= O(eπt4 ). (9.25)

This discussion shows that, while our asymptotic estimates for f , φ, and V , aregood for low values of x, x < T0, one cannot use them for large values of x exceptin estimating integrals as in Equation (9.21).

The well established results for a unitary S and real potentials now guaranteethat Schrödinger’s equation holds for f ±(it; k; x) and U(it; x), i.e.

− d2

dx2f ±(it; k; x) + U(it; x)f ± = k2f ±, t > T0. (9.26)

We are also guaranteed that

f ±(it; k; 0) = M(±)(it; k), (9.27)

where M(±) is the original Jost function we started with

M(±)(it; k) = 1 − (t2 + 1/4)∫ ∞

1

ψ(α)α1/4[αit/2 + α−it/2]α ± ik

dα. (9.28)

Thus by analytic continuation the Jost solutions f ±(ν; k; x) given in the previoussections will also give the original Jost function, i.e. (9.28) with it replaced by ν,and ν ∈ S(T0).

10. Asymptotic Expansion in Powers of g

In this section, we carry out the asymptotic expansion of the kernels F and A andthe potential V in powers of g, where g ≡ (ν2 − 1/4)−1, and ν ∈ S(T0) withT0 > 103.

We start with the definition of the Marchenko kernel F(ν; x) given in Equa-tion (8.2),

F(ν; x) = 1

2π

∫L

dk[S(ν, k) − 1]eikx , x > 0, (10.1)

where L is the line Im k = 1/4.Using the asymptotic expansion of M(±)(ν, k) given in Equation (5.47), we have

S(ν, k) − 1 =∑N

n=0gnM(+)

n (k) + gNR(+)N (ν, k)∑N

n=0gnM(−)

n (k) + gNR(−)N (ν, k)

− 1, (10.2)

where R(±)N (ν, k) are both O(g) for small g.


Next we recall the lower bound obtained in Section 8 for M(−)0 (k) when k =

λ + i(1/4), and −∞ < λ < +∞. This is given in Equation (8.30)

|M(−)0 (k)| � 0.255, k ∈ L. (10.3)

From Lemma 6.1, we get also a lower bound on |M(−)(ν, k)| for ν ∈ S(T0), andk ∈ L given in Equation (8.31)

|M(−)(ν, k)| � 1/4, T0 > 103. (10.4)

The last two bounds guarantee that the denominator in Equation (10.2) does notvanish for any k ∈ L and ν ∈ S(T0). We can then proceed to expand [S(ν, k) − 1]in powers of g for any k ∈ L and get

[S(ν, k) − 1] =N∑n=0

gnHn(k) + gNH(N)R (g, k), (10.5)

where from Equation (10.2) we get

H0(k) ≡ S0(k) − 1 = M(+)0 (k)

M(−)0 (k)

− 1, (10.6)

H1(k) ≡ 1

M(−)0 (k)

{M

(+)

1 (k) − M(+)0 (k)M

(−)1 (k)

M(−)0 (k)

}(10.7)

and

H2(k) ≡ 1

M(−)0 (k)

{M

(+)2 (k) − M

(+)

1 (k)M(−)

1 (k)

M(−)0 (k)

− M(−)2 (k)M

(+)0 (k)

M(−)0 (k)

+

+ M(+)0 (k)[M(−)

1 (k)]2

[M(−)0 (k)]2

}(10.8)

with similar expressions for Hn(k), n > 2, which we will not need in this paper.The remainder term H

(N)R is O(g) as g → 0. All Hn(k) are rational functions of k.

Equation (10.5) immediately gives us the asymptotic expansion for the kernelF(ν; x),

F(ν; x) = F0(x) + gF1(x) + g2F2(x) + · · · + gNF(N)R (g; x), (10.9)

where

Fn(x) = 1

2π

∫L

dkHn(k)eikx, x � 0. (10.10)

This last Fourier transform is conditionally convergent since |Hn(k)| = O(1/k)for large k (note that L is the line Im k = 1/4).

46 N. N. KHURI

Next we stress that all the Hn(k) are analytic for 1 > Im k > 0. The denomina-tors in Equations (10.6)–(10.8), do not vanish in Im k � 0, except at k = 0.

This also holds for H(N)R (g, k). Thus we can shift back the contour in Equa-

tion (10.10) to the real k-axis and obtain

Fn(x) = 1

2π

∫ +∞

−∞dkHn(k)e

ikx, x � 0. (10.11)

For real k, it follows from Equation (5.49) that [M(+)n (k)]∗ = M(−)(k), and that

M(+)n (−k) = M(−)

n (k). This leads us to

H ∗n (k) = Hn(−k), for k real. (10.12)

Thus it immediately follows from Equation (10.11) that all Fn(x), n = 0, 1, 2,. . . , N, are real functions. However, F (N)

R (g, x), is certainly not real for ν ∈ S(T0).We have explicitly calculated F0(x) in Section 7, and obtained

F0(x) = λ0e−x + λ1xe−x , (10.13)

with λ0 and λ1 real and given in Equations (7.6) and (7.7).One can also easily calculate explicitly F1(x) by contour integration.

F1(x) = 1

2π

∫ +∞

−∞dk

[M

(+)

1 (k)

M(−)0 (k)

− M(+)0 (k)M

(−)1 (k)

[M(−)0 (k)]2

]eikx. (10.14)

The result is

F1(x) =(

3∑n=0

σnxn

)e−x, (10.15)

where the constants σn are explicitly given as functions of a1, and bj , j = 1, . . . , 4.Here we will only give the numerical value of the σ ′

ns

σ0 = 26.5228, σ1 = 1.7901,(10.16)

σ2 = −9.3291, σ3 = −2.3582.

The result for F2(x) will be similar,

F2(x) ≡ e−x

(5∑

n=0

βnxn

). (10.17)

We will not give its explicit value as it is not needed.The Marchenko equation (8.10), for ν ∈ S(T0) can be written in operator

form

A = F + AF. (10.18)


Writing

A = A0 + gA1 + g2A2 + g2A(2)R , (10.19)

and using the expansion for F given in Equation (10.9) we get by comparing terms

A0 = F0 + A0F0, (10.20)

an equation we solved explicitly in Section 7. In addition, we have

A1 = (F1 + A0F1) + A1F0, (10.21)

as the integral equation for A1, and

A2 = (F2 + A1F1 + A0F2) + A2F0, (10.22)

for A2. It is obvious that the integral equations for An, n = 0, 1, 2, . . . all have thesame kernel F0. Thus, they are all explicitly solvable. Given our solution, A0, forEquation (10.20), we get

A0 = (1 − F0)−1F0 (10.23)

and, hence,

(1 − F0)−1 = A0 + 1. (10.24)

This leads to solutions for A1, A2, etc., with

A1 = F1 + 2A0F1 + A0(A0F1) (10.25)

and, given A1, we can now get A2 explicitly as

A2 = F2 + A1F1 + 2A0F2 + A0(A1F1) + A0(A0F2). (10.26)

It is now obvious that all the An’s are real and continuously differentiable, fory � x � 0, since from Equation (7.11),

A0(x, y) = [B(x) + (y − x)C(x)]e−(y−x), (10.27)

with B(x) and C(x) given by Equations (7.18) and (7.19) and B,C are O(e−2x) asx → ∞. The kernels Fn(x) are of the form

Fn(x) = e−x

[2n+1∑j=0

σ(n)j xj

]. (10.28)

The potential U(ν; x) is given by

U(ν; x) = −2d

dxA(ν; x, x), x � 0. (10.29)

48 N. N. KHURI

Figure 1. Plot of V1(x).

Using the variable g ≡ (ν2 − 1/4)−1, we write

U(ν; x) ≡ V (g; x). (10.30)

The expansion of A in powers of g, given in Equation (10.19), now gives us,

V (g; x) = V0(x) + gV1(x) + g2V2(x) + · · · + gNV(N)R (g, x), (10.31)

where

Vn(x) = −2d

dxAn(ν; x, x), (10.32)

V(N)R (g, x) = −2

d

dxA

(N)R (ν; x, x) (10.33)

and V(N)R is O(g).

All the Vn’s are real, continuous for x ∈ [0,∞), and O(e−2x) for large x.V

(N)R (g, x) is complex for ν ∈ S(T0) but also continuous and O(e−2x) for large x.

It is now clear why one can refer to ‘g’ as a coupling constant specially for largevalues of Im ν, i.e., small g, g = O(|Im ν|−2).

We have already calculated the first term in the expansion, V0(x), and it is givenexplicitly in Equation (7.21). Later, we will need to have V1(x) and we proceed tocalculate it here.

From Equation (10.25), we get

A1(x, x) = F1(2x) + 2∫ ∞

x

dzA0(x, z)F1(z + x) +

+∫ ∞

x

dz1

∫ ∞

x

dz2A0(x; z1)A0(x; z2)F1(z1 + z2), (10.34)

where A0 and F1 are given in Equations (10.27) and (10.25). It is now evident thatA1(x, x) is continuously differentiable and

V1(x) = −2∂A1

∂x(x, x), (10.35)


can be explicitly calculated in terms of B(x), C(x), and the real constants σj . Theresulting V1(x) is continuous, finite at x = 0, and V1 = O(x3e−2x) for large x. Wedo not write down the full result, but exhibit a graph of V1(x) in Figure 1.

In closing we comment on the asymptotic expansion (10.31) for V (g; x). It isnot, of course, convergent no matter how small |g| is. This follows from the fact thatthe expansions for M±(ν; k) are also divergent. There is an essential singularity atg = 0. The constants χ(n)

5 , 5 = 1, . . . , 2n+ 2, given in Equations (5.32) and (5.33)grow fast. However, Equation (10.31) can still give an extremely good estimatefor V (g; x) as long as N is O(1). Indeed it is possible to get a uniform bound onV

(N)R (g, x) which is

|V (N)R (g, x)| � C(N)

|Im ν|2 e−x, x > 0, (10.36)

where C(N) grows fast with N . For the purposes of this paper we need at mostN = 2 or 3. For T0 = 104, (C(2)/T 2

0 ) � 1. This makes (g2V(2)R ) smaller than

(O(1)/|Im ν|4)e−x , and thus V0+gV1+g2V2, give an excellent estimate of V (g, x)

for all ν ∈ S(T0), and x not large. However, this estimate is not good for large x

where both V and the error are small.Finally, we will need an important result on the phase of V (N)

R near the criticalline ν = it .

As we have shown in Section 9, F(it; x), A(it; x; y), and U(it; x) are all real,for x ∈ [0,∞) and y � x. In addition, in the asymptotic expansion given in Equa-tion (10.31) all the coefficients Vn(x) are real. But the remainder term, V (N)

R (g, x),is in general complex for ν = ω + it , and ω �= 0. However, for ω = 0, we againhave reality

U(N)R (it; x) ≡ V

(N)R (−(t2 + 1

4 )−1; x) = (V

(N)R )∗. (10.37)

In Section 8 we proved that both A(ν; x, y) and U(ν; x) are analytic in ν forν ∈ S(T0). Hence, so is U(N)

R (ν; x). This leads us to the following lemma:

LEMMA 10.1. For ν ∈ S(T0), and ν = δ + it , we have

|Im(V(N)R (g; x))| = |ImU

(N)R (δ + it; x)| < C(x) · |δ| + O(δ2), (10.38)

where

C(x) � c1

t2e−x. (10.39)

Proof.

[ImU(N)R (ν; x)] = Im

dU(N)R (ν; x)

dν

∣∣∣∣ν=it

(ν′ − ν) + O((ν′ − ν)2). (10.40)

The derivative is finite and setting ν′ = δ + it , δ � 1/2, we get Equation (10.38).

50 N. N. KHURI

The bound on C(x) follows from our previous estimates. We will not give theproof here. ✷

11. The Zeros of M(−)(ν;k) for Fixed k

In this section we shall study the properties of the infinite set of zeros, νn(k), ofM(−)(ν, k) for fixed k. We prove three lemmas for {νn(k)}.

For convenience and without loss of generality, we set k = iτ , and τ � 0. Wewrite

M(ν; τ) ≡ M(−)(ν, iτ ) (11.1)

and

Mn(τ) ≡ M(−)n (iτ ). (11.2)

It is clear from the equation defining M(ν, τ), i.e. Equation (3.10) with k =iτ , that M(ν, τ), fixed τ > 0, is an entire function of ν with order the same asξ(ν + 1/2), i.e. order 1. Hence, M(ν, τ) has an infinite set of zeros, νn(τ), with|νn(τ)| → ∞ as n → ∞.

Next it follows from Section 6 that, νn(τ), will all be outside the truncated

critical strip, S(T0), for τ > T N0 e− πT0

4 . As we decrease τ , νn(τ), will start appearing

in S(T0) for τ < O(T pe− πT4 ).

Our first lemma is the following:

LEMMA 11.1. As τ → 0,

limτ→0

νn(τ) = νn, (11.3)

where (νn + 1/2) is a zero of the zeta function, i.e.

ξ(νn + 12) = 0. (11.4)

Proof. From the asymptotic expansion of M(ν; τ) given in Equation (5.47), wehave

M(ν; τ) = 2ξ(ν + 12 ) + M0(τ ) + gM1(τ ) + g2M2(τ ) + g2R2(ν, τ ), (11.5)

where

R2(ν, τ ) =∫ ∞

1dα

(7∑

5=1

6(3)5 (α)

{1

(α + τ)5− 1

α5

})[αν/2 + α−ν/2]. (11.6)

In getting Equations (11.4) and (11.5), we have used Equation (5.46) for the ξ -function.


By definition, we have

M(νn(τ), τ ) ≡ 0. (11.7)

Hence, we get

−2ξ(νn(τ) + 12 ) = M0(τ ) + gn(τ)M1(τ ) + g2

n(τ )M2(τ ) ++ g2

n(τ )R2(νn(τ); τ), (11.8)

with

gn(τ) = 1

[ν2n(τ ) − 1

4 ] . (11.9)

Now all the terms on the right-hand side of Equation (11.8) are O(τ ) as τ → 0.Hence, we get

limτ→0

ξ(νn(τ) + 12 ) = ξ(νn(0) + 1

2) = 0, (11.10)

and, therefore,

νn(0) = νn. ✷The next lemma gives us as estimate of (νn(τ) − νn(0)) as τ → 0.

LEMMA 11.2. If νn is a first-order zero of ξ(ν + 1/2), then as τ → 0

(νn(τ) − νn) = O(τ ), (11.11)

and if νn is of order p, then

(νn(τ) − νn) = O(τ 1/p). (11.12)

Proof. Since ξ(ν + 1/2) is entire we can write for any νn

ξ(νn(τ) + 12 )

= ξ(νn + 12 ) + dξ

dν

∣∣∣∣ν=νn

(νn(τ) − νn) + O[(νn(τ) − νn)2]. (11.13)

But the first term on the right is zero, and (ξ ′)ν=νn �= 0 for a first-order zero, andfrom Equation (11.7) we get

−2(ξ ′)ν=νn(νn(τ) − νn) = τ [(2 + a1) + O(gn(τ))] + O(τ 2). (11.14)

This gives our proof for a first-order zero. ✷For a zero of multiplicity p, we have by definition (ξ (j))ν=νn = 0, for j =

1, 2, . . . , p − 1, and (ξ (p))ν=νn �= 0. Hence, we get

−2[ξ (p)]ν=νn(νn(τ) − ν)p = τ [(2 + a1) + O(gn(τ))] + O(t2). (11.15)

52 N. N. KHURI

This leads to

(νn(τ) − νn) = O(τ 1/p), τ → 0. (11.16)

So far we have shown that every νn(τ) approaches a Riemann zero as τ → 0,but have not established the converse, i.e. that any νn is the limit of a νn(τ) asτ → 0.

To do this we first define a rectangular region R(T0, T ) as follows:

R(T0, T ) = {ν | −3/2 � Re ν � 3/2;T0 � Im ν � T }, (11.17)

with T � T0.We now prove our third lemma.

LEMMA 11.3. Let Nξ(T0, T ) be the number of zeros of ξ(ν + 1/2) for ν ∈R(T0, T ), and NM(T0, T ; τ) be the number of zeros, νn(τ), of M(ν, τ), with νn(τ) ∈R(T0, T ), then for sufficiently small τ ,

|NM(T0, T ; τ) − Nξ(T0, T )| < 1. (11.18)

There exists a small interval in τ , 0 � τ � τ0(T ), such that

NM(T0, T ; τ) = Nξ(T0, T ). (11.19)

Proof. We start with the standard expression:

NM − Nξ = 1

2πi

∮+R

dν

{M ′(ν; τ)M(ν; τ) − ξ ′(ν + 1/2)

ξ(ν + 1/2)

}, (11.20)

where +R is the boundary of the rectangle R. We also choose T and T0, suchthat they both lie between the abscissa of successive zeros νn, i.e. Im νn1 < T <

Im νN1+1, and Im νn0 < T0 < Im νn0+1. Thus, +B never has a zero of ξ on it. Theprime in (11.20) denotes (d/dν). We follow the method used to prove theorem 9.3in [18].

Using the symmetry of ξ in ν, we have

Nξ = 1

πi

∫ 32 +iT

32 +iT0

dνξ ′

ξ+ 1

πi

∫ iT

32 +iT

dνξ ′

ξ+ 1

πi

∫ 32 +iT0

iT0

dνξ ′

ξ

= 1

π{� arg ξ(ν + 1

2)}, (11.21)

where E denotes the variation from iT0 to 3/2 + iT0 then from (3/2 + iT0) to3/2 + iT , and thence to iT .

But from Equation (11.5), we get

{M(ν; τ)/ξ(ν + 12)} = 1 + (2 + a1)τ + cgτ

2ξ(ν + 12)

+ O(τ 2). (11.22)


On the horizontal parts of +R , we can use Theorem 9.7 of [18] to obtain alower bound on |ξ(ν + 1/2)|. Indeed, there is a constant, A, such that each interval(T , T + 1) contains a value of t for which

|ζ(ν + 12 )| > t−A, − 3

2 � ω � 32 , (11.23)

where ν = ω + it . On the vertical parts of +B ,|ζ(ν + 1/2)| is obviously boundedfrom below. Using the standard asymptotic expression for +(1/4+ν/2) as t → ∞,we finally get

|[argM(ν; τ) − arg ξ(ν + 12 )]| � τ

|ξ(ν + 12)|

. (11.24)

We can choose τ = T −A−N(e−πT

4 ) with N � 3 and obtain

E(argM(ν, τ) − arg ξ(ν + 12 )) � 1

T N−1. (11.25)

Hence, as T → ∞, we get

NM − Nξ = 0. (11.26)

This completes our proof. ✷We stress one important fact that is a consequence of the results of this section.

Namely, we are now not limited to the study of the zero energy zeros, νn(0). Onecan consider the case for small enough τ , but τ > 0, and obviously if there is aninterval 0 < τ < τn for which [Re νn(τ)] = 0, this will be sufficient for the validityof the Riemann hypothesis. In the next section we will see the importance of thisremark.

12. The Potential V (g, x) and the Riemann Hypothesis

Following the notation of the previous section, we define the Jost solution f (g; τ, x)as

f (g; τ ; x) = f (−)(g; iτ, x), (12.1)

with k = iτ , and ν ∈ S(T0). We also recall the result given in Faddeev’s review[14]

|f − e−τx | � Ke−τx

∫ ∞

x

u|V (g, u)| du, (12.2)

hence, f = O(e−τx) as x → ∞, τ � 0.We write the Schrödinger equation for f and f ∗,

−d2f

dx2+ V (g, x)f = −τ 2f (12.3)

54 N. N. KHURI

and

−d2f ∗

dx2+ V ∗(g, x)f ∗ = −τ 2f ∗. (12.4)

Multiplying the first equation by f ∗ and the second by f , integrating from x = 0to x = ∞, and subtracting the two equations, we get

2i∫ ∞

0dx|f (g; τ ; x)|2[ImV (g, x)]

= M(ν, τ)K∗(ν, τ ) − M∗(ν, τ )K(ν, τ), (12.5)

where

K(ν, τ) =[

df (g; τ ; x)dx

]x=0

, (12.6)

and g = (ν2 − 1/4)−1. The derivative exists for x → 0 in our present case, sinceV (g, 0) is finite. One can also check this from the expression for f in terms ofA(ν; x, y),

f (g; τ, x) = e−τx +∫ ∞

x

dyA(ν; x, y)e−τy . (12.7)

From Section 8, we know that A(ν; 0, 0) is finite, and also (∂A/∂x), for x � 0,y � x exists and is integrable.

Next we set ν = νn(τ), and g = gn(τ) in Equation (12.3), and we get∫ ∞

0dx|f (gn(τ); τ ; x)|2[ImV (gn(τ); x)] = 0. (12.8)

This last integral is absolutely and uniformly convergent, since V = O(e−2x) asx → ∞. Thus, we can take the limit τ → 0 and obtain∫ ∞

0dx|f (gn; 0; x)|2[ImV (gn; x)] = 0. (12.9)

Now V (g, x) has an asymptotic expansion in g.

V (g; x) = V0(x) + gV1(x) + g2V2(x) + g2V(2)R (g, x). (12.10)

Also from the expansion for A(ν; x, y), for the Jost solution we get

f (g; τ ; x) = f0(τ ; x) + gf1(τ ; x) + g2f2(τ ; x) + g2fR(g; τ ; x). (12.11)

Both V(2)R and f

(2)R are O(g) and have bounds in x. It is sufficient, for the validity

of the Riemann hypothesis for a constant c > 0, c = O(1), to exist such that∫ ∞

0dx|f0(x)|2V1(x) = c > 0, (12.12)


where f0(x) = f0(0, x).To prove this last statement, we first note that setting νn = νn(0), we have

νn = ωn + itn, tn > T0 (12.13)

and

Im gn = 2ωntn

[t2n − ω2

n + 1/4]2, ω2

n � 14 . (12.14)

The vanishing of Im gn with tn > T0 > 0, implies ωn = 0 and, hence, sn =1/2 + itn.

Next, from Equation (12.10), we write

ImV (gn; x)= (Im gn)V1(x) + (Im g2

n)V2(x)++ (Im g2

n)(ReV (2)R (gn; x)) + (Re g2

n)(ImV(2)R (gn; x)). (12.15)

Also in the integrand in (12.0), we can write

|f (gn; 0; x)|2 = |f (0; 0; x)|2 + O

(1

t2n

). (12.16)

In Equation (12.15) we note that

(Im gn) = O

(ωn

t3n

)and Im g2

n = O

(ωn

t5n

),

while (Re g2N) = O(1/t4

n). On substituting Equations (12.15) and (12.16) in (12.9),and using the assumption (12.12), we see at first that consistency requires ωn tobe small, i.e. ωn = O(1/t3

n). Here we use the fact that |V (2)R | = O(g) for small g.

However, we have more information on V(2)R and specifically its phase for small ωn.

This was given in Lemma 10.1, where it was shown that ImV(2)R = O(ω) for small

ω and that V (2)R is real for ω = 0. Given the bound on ImV

(2)R from this lemma, we

see that the leading contribution from (12.15) to Equation (12.9) must come from[(Im gn)V1] and cannot be cancelled by the other three terms. We have

(Im gn)c + (Im g2n)X1 + (Im g2

n)X2 + (Re g2n)X3 = 0, (12.17)

with

|X1| � 2∫ ∞

0|f0(x)|2|V2(x)| dx,

|X2| � 2∫ ∞

0|f0(x)|2| ReV (2)

R (gn; x)|,

|X3| � 2∫ ∞

0|f0(x)|2| ImV

(2)R (gn, x)| dx

� 2ωnc1

t2n

∫ ∞

0|f0(x)|2e−x dx. (12.18)

56 N. N. KHURI

Hence we have constants Bj such that |Xj | < Bj ; j = 1, 2, and |X3| <

(ωn/t2n)B3. Note that in the last inequality we used Lemma 10.1. We take T0 large

enough such that |Bj |/T 20 � c, j = 1, 2, 3; and with c = O(1). Next we rewrite

Equation (12.17) as

(Im gn)[c + 2(Re gn)X1 + 2(Re gn)X2 + (Re g2n) · tnX3] = 0, (12.19)

where now |X3| < 2B3. The term in the square bracket cannot vanish and, hence,we obtain

[Im gn]c = [Im gn]∫ ∞

0|f0(x)|2V1(x) dx = 0, (12.20)

and thus, if c �= 0, then for all tn > T0, we get

Im gn = 0 or sn = 12 + itn. (12.21)

We have already calculated f0(x) exactly in Section 7, and V1(x) in Section 10,and we can compute the integral in (12.12) directly, the result is∫ ∞

0|f0(x)|2V1(x) = 0. (12.22)

This can be checked numerically, and indeed can be rigorously proved. So we haveno information on (Im gn) from (12.17). However, the proof of (12.22) suggests tous how we can proceed further.

To prove (12.22) we use the Schrödinger equation and the expansions (12.10)and (12.11). We obtain

−d2f0

dx2(τ, x) + V0(x)f0(τ, x) = −τ 2f0(τ, x) (12.23)

and

−d2f1

dx2(τ, x) + V0(x)f1(τ, x) + V1(x)f0(τ, x) = −τ 2f1(τ, x). (12.24)

Multiplying the first equation above by f1 and the second by f0, integrating fromzero to infinity, and subtracting, we have∫ ∞

0[f0(τ, x)]2V1(x) dx = −[f1(τ, 0)f ′

0(τ, 0) − f0(τ, 0)f ′1(τ, 0)], (12.25)

where the prime denotes (d/dx).By definition, we have

f0(τ, 0) = M0(τ ) (12.26)

and

f1(τ, 0) = M1(τ ). (12.27)


Both M0(τ ) and M1(τ ) are O(τ ) for small τ and vanish as τ → 0. We then get,after taking the limit,∫ ∞

0|f0(0, x)|2V1(x) dx ≡ 0. (12.28)

The crucial factor here is the fact that M(ν, 0) = 2ξ(ν + 1/2), and for large |Im ν|,ν ∈ S(T0),

M(ν, 0) = 2ξ(ν + 12) = O(e

−π | Im ν|4 ).

This forces all the coefficients, Mn(τ), in the asymptotic expansion of M(ν, τ) inpowers of g to vanish as τ → 0. The culprit is the factor +(ν/2 + 1/4) in theEquation (3.4) which relates ξ(s) to ζ(s). We will also see below how this facthinders us in treating the case τ �= 0, but τ small.

From the results of Section 11, it is evident that it is sufficient to prove thatRe νn(τ) = 0 in an interval 0 < τ < τ0(n), where

τ0(n) = O

(exp

(−π | Im νn|4

)).

From Equation (12.19), one can prove that∫ ∞

0[f0(τ, x)]2V1(x) dx = Kτ + O(τ 2), (12.29)

where K is a constant, K = O(1). The integral does not vanish if τ > 0.This suggests trying a double expansion in powers of g and τ . However, again

this will not lead to any restriction on Im gn. The main problem is the relevantdomain in τ is small, i.e. τ = O(e

−πt4 ), and terms of order g2 are much larger than

terms of order τ .To proceed further along the lines suggested by this paper, one has to do two

things:(i) First, find an even function h(ν), analytic for ν ∈ S(T0), and having no zeros

in S(T0), such that if we define, χ(ν) as

χ(ν) ≡ ξ(ν + 12 )h(ν), (12.30)

we have

χ(ν) = O([t2]−p), 1 < p < 32 , (12.31)

where ν = ω+ it , t > T0. The point here is that χ(ν) is small but not smaller thanO(g2).

This first step is achievable. For example, we can define χ as

χ(ν) ≡ ξ(ν + 12 )[cos π

4 ν](ν2 − 1

4 )2+δ

, 14 > δ > 0. (12.32)

58 N. N. KHURI

This will give

χ(ν) = O([t2]−(1+δ)). (12.33)

The second requirement is much harder to achieve:(ii) One has to construct Jost functions, M(±)

χ (ν, k), preferably of the Martintype, such that

limk→0

M(ν)χ = χ(ν + 1

2) (12.34)

and for small g, ν ∈ S(T0).In addition, M±

χ has to be of the Martin type and it must have an asymptoticexpansion in powers of g = (ν2 − 1/2)−1.

Appendix

In this Appendix we first give a proof of the Laplace transform representation forthe Marchenko kernel, F(ν; x), x > 0.

Starting with the definition (8.2)

F(ν; x) = 1

2π

∫L

(S(ν; k) − 1)eikx dk, (A.1)

where L is the line Im k = δ, with 1/4 � δ < 1, we note that for ν ∈ S(T0) we

have S(ν; k) analytic in k for Im k > δ > O(e−πT0

4 ), except for the cut along thepositive imaginary k-axis, 1 � Im k < ∞. Second, in this region, we have a boundfor large |k|

|S(ν, k) − 1| < C

|k| , |k| → ∞. (A.2)

This bound holds along any radial direction that excludes the cut.We can deform the contour L from along the line Im k = 1/4, to a contour

surrounding the cut, i.e.

F(ν; x) = 1

2π

∫C

[S(ν; k) − 1]eikx dk, x > 0, (A.3)

where C starts at (−ε,+i∞) and descends to (−ε,+i), turns around the point k =i, and then extends from (+ε,+i) to (+ε,+i∞). The contribution from the largesemicircle, |k| = K, vanishes as K → ∞. For |K|−1/2 < arg k < π −|K|−1/2, thecontribution is O(exp(−|K|1/2x)), and vanishes for x > 0. Here we use the bound(A.2). For the regions

0 � arg k � |K|−1/2, and π − |K|−1/2 � arg k � π,

the contribution from the semicircle to (A.1) will be O(|K|−1/2) and also vanishesas |K| → ∞.


From (A.3), we finally obtain

F(ν; x) = 1

π

∫ ∞

1D(ν;α)e−αx da, x > 0, (A.4)

where D is given in Equation (8.5), and from Equation (4.2) is just the discontinuityof S(ν; k) along the cut. Noting that D = O(e−πα) as α → ∞, we see that (A.4)will also hold for x = 0.

The next task for this Appendix is to give a direct proof of the fact that U(ν; x)and f (±)(ν; k; x) as defined in Equations (8.93) and (8.94) do indeed satisfy theSchrödinger equation, i.e. to prove Lemma 8.5 directly. We will also give an ex-plicit expression for U(ν; x).

Following [21], we define an operator, Q(ν; x), depending on two parameters,ν and x, with Re x � 0, and ν ∈ S(T0). Q acts on functions u(β), 1 � β < ∞,with u ∈ L2(1,∞). We define Q as

[Q(ν; x)u](α) ≡∫ ∞

1Q(ν; x;α, β)u(β) dβ, (A.5)

where

Q(ν; x;α, β) = 1

π

D(ν;β)e−2βx

[α + β] , Re x � 0, (A.6)

with D(ν;β) given by Equations (4.3) and (4.4). Q will have a finite Hilbert–Schmidt norm

Q(ν; x) � Ke−2x, (A.7)

where K depends on ν.We introduce a new integral equation,

W(ν; x;α) = 1 +∫ ∞

1Q(ν; x;α, β)W(ν; x;β) dβ, (A.8)

with W ∈ L2(1,∞).This integral equation is equivalent to the Marchenko equation (8.10). To see

that, we write

Z(ν; x;α) ≡ 1

πD(ν;α)e−αxW(ν; x;α). (A.9)

A(ν; x, y) is

A(ν; x, y) ≡∫ ∞

1Z(ν; x;α)e−αy dα. (A.10)

The Laplace transform exists since D = O(e−πα) and W ∈ L2. Substituting (A.9)and (A.10) in Equation (A.8) and using (A.4), we obtain for A

A(ν; x, y) = F(ν; x + y) +∫ ∞

x

A(ν; x, u)F (ν;u + y)a. (A.11)

60 N. N. KHURI

But this is just the Marchenko equation which we have shown in Section 8 doeshave a unique solution A. Hence, A = A for all ν ∈ S(T0). Thus we concludethat Equation (A.8), which is of the Fredholm type, has a unique solution, sincethe homogeneous equation W = QW cannot have a solution for that will lead tothe existence of a solution for A = AF which we have shown in Section 8 is notpossible.

Given the function W(ν; x;α), we can easily get expressions for U(ν; x) andf ±(ν; k; x).

The potential is given by

U(ν; x) = −2d

dx

[1

π

∫ ∞

1D(ν, α)e−2αxW(ν; x;α) da

], Re x � 0. (A.12)

Similarly, from (A.9) and (A.10), we get

f (±)(ν; k, x) = e∓ikx + e∓ikx

(1

π

)∫ ∞

1

D(ν;α)e−2αxW(ν; x;α)(α ± ik)

dα. (A.13)

To check that we recover the same Jost functions we started with, we write

f (±)(ν; k; 0) ≡ M(±)(ν; k) ≡ 1 +(

1

π

)∫ ∞

1

D(ν;α)W(ν; 0;α)(α ± ik)

dα. (A.14)

However, the integral equation for W for x = 0 is trivially soluble. From Equa-tion (A.8), we have

W(ν; 0;α) = 1 + 1

π

∫D(ν;β)W(ν; 0;β)

(α + β)dβ. (A.15)

Setting

W(ν; 0;α) ≡ M(−)(ν; iα) (A.16)

and using Equations (4.3) and (4.4) for D(ν;α), we have

M(−)(ν; iα) = 1 + (ν2 − 14 )

∫ ∞

1dβ

ψ(β)β1/4[βν/2 + β−ν/2]β + α

. (A.17)

This is our original expression for M(−).The operator Q is a Fredholm-type operator, and for Re x � 0, and ν ∈ S(T0),

we proved that there are no nontrivial solutions of the homogeneous equation u =Qu. Hence, the determinant, Det(1 − Q), cannot vanish for any Re x � 0. Thisdeterminant can be calculated explicitly, as was done in [21], and we obtain

Det(1 − Q(ν; x)) = 1 +∞∑n=1

1

(n!)∫ ∞

1dα1 . . .

∫ ∞

1dαn

(n∏

j=1

D(ν, αj )e−2αj x

2παj

)×

×n∏

i<j

(αi − αj)2

(αi + αj)2. (A.18)


This series is absolutely convergent for all Re x � 0, ν ∈ S(T0), and | Im ν| < ∞.Also as shown in [21], the potential U(ν; x) is now given by Dyson’s [23] formula,i.e.

U(ν; x) = −2d2

dx2{log[Det(1 − Q(ν; x))]}. (A.19)

Finally, we give here a direct check on the validity of the Schrödinger equationfor f ± and U(ν; x).

From Equation (A.7) giving a bound on the norm of Q, we see that for some x0,Q < 1 for all (Re x) > x0, x0 � logK/2. Thus the iterative series for W(ν; x;α)is absolutely convergent for all Re x > x0,

W = 1 +∞∑n=1

Qn. (A.20)

Using Equations (A.6), (A.12), and (A.13), we obtain

U(ν; x) = 4∞∑n=0

(1

π

)n+1 ∫ ∞

1dα0 . . .

∫ ∞

1dαn ×

×∏n

j=0 D(ν;αj)e−2αj x∏n−1j=0(αj + αj+1)

(n∑

j=0

αj

), Re x > x0. (A.21)

A similar series holds for f ±:

f (±)(ν; k; x) = e∓ikx + e∓ikx

∞∑n=0

(1

π

)n+1 ∫ ∞

1dα0 . . .

∫ ∞

1dαn ×

×∏n−1

j=0 D(ν;αj)e−2αj x

[∏n−1j=0(αj + αj+1)](α0 ± ik)

. (A.22)

Using these series, we can check directly for Re x > x0, that U and f ± give apotential and its unique Jost solutions.

We define h(±):

h(±) = e±ikxf (±). (A.23)

The Schrödinger equation for h(±) is now

d2

dx2h±(ν; k; x) ∓ 2ik

dh±

dx(ν; k; x) = U(ν; x)h±(ν; k; x). (A.24)

Substituting, expressions (A.21) and (A.22) in the above, we see, after some alge-bra, that for x > x0, (A.24) is satisfied if the following algebraic identity holds[

r−1∑n=0

{(αn + αn+1)

(n∑

j=0

αj

)}]=(

n∑j=0

αj

)2

− αr

(r∑

j=0

αj

). (A.25)

62 N. N. KHURI

But this is equivalent to

r∑n=0

αn

(n∑

j=0

αj

)+

r−1∑n=0

(αn+1)

(n∑

j=0

αj

)=[

r∑j=0

αj

]2

. (A.26)

This last equation is an identity and can be proved by induction.The Schrödinger equation is thus valid for all x > x0. But again using analytic

continuation, now in x we easily see that it must hold for all x � 0. All the termsin (A.24) are analytic in the half plane Re x � 0.

Acknowledgements

The author wishes to thank James Liu and H. C. Ren for untiring help in checkingmuch of the algebraic manipulations in this paper including the use of Mathe-matica to produce Tables I and II and carry out other numerical work. This workwas supported in part by the U.S. Department of Energy under grant numberDOE91ER40651 TaskB.

References

1. Dyson, F. J.: J. Math. Phys. 3 (1962), 140.2. Montgomery, H. L.: In: Proc. Sympos. Pure Math. 24, Amer. Math. Soc., Providence, RI, 1973,

pp. 181–193.3. Berry, M. V.: ‘Riemann’s Zeta Function: a Model of Quantum Chaos,’ Lecture Notes in Phys.

262, Springer, New York, 1986.4. Chadan, K.: private communication, see also K. Chadan and M. Musette, C.R. Acad. Sci.

Paris (2) 316 (1993), 1–6. In this paper an example is given with some important propertiesof the zeta function demonstrated.

5. Meetz, K.: J. Math. Phys. 3 (1962), 690.6. Gelfand, I. M. and Levitan, B. M.: Izvest. Akad. Nauk. SSSR Ser. Matem. 15 (1951), 309.7. Marchenko, V. A.: Dokl. Akad. Nauk SSSR 104 (1955), 695. [Math. Rev. 17 (1956), 740].8. Martin, A.: Nuovo Cimento 19 (1961), 1257.9. Jost, R.: Helv. Physica Acta 20 (1947), 256.

10. Levinson, N.: Kgl. Danske Videnskab. Selskab, Math.-fys. Medd. 25(9) (1949).11. Bargmann, V.: Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 961.12. Jost, R. and Pais, A.: Phys. Rev. 82 (1951), 840.13. Chadan, K. and Sabatier, R. C.: Inverse Problems in Quantum Scattering Theory, 2nd edn,

Springer, New York, 1989.14. Faddeev, L. D.: J. Math. Phys. 4 (1963), 72.15. Blankenbecler, R., Goldberger, M. L., Khuri, N. N. and Treiman, S. B.: Ann. of Phys. 10 (1960),

62.16. Regge, T.: Nuovo Cimento 14(5) (1959), 951.17. Martin, A.: Nuovo Cimento 14 (1959), 403.18. Titchmarsh, E. C.: The Theory of the Riemann Zeta-function, 2nd edn, revised by D. R. Heath-

Brown, Oxford Univ. Press, Oxford, 1986.19. Gross, D. J. and Kayser, B. J.: Phys. Rev. 152 (1966), 1441.20. Cornille, H.: J. Math. Phys. 8 (1967), 2268.


21. Khuri, N. N.: Inverse scattering revisited: explicit solution of the Marchenko–Martin method,In: S. Ciulli, F. Scheck and W. Thirring (eds), Rigorous Methods in Particle Physics, Springer-Verlag, Berlin, 1990, pp. 77–97.

22. Bargmann, V.: Rev. Modern Phys. 21 (1949), 488.23. Dyson, F. J.: In: E. Lieb, B. Simon and A. S. Wightman (eds), Studies in Mathematical Physics,

Princeton Univ. Press, Princeton, NJ, 1976, pp. 151–167.


65

Algebras of Operators on HolomorphicFunctions and Applications

M. BEN CHROUDA and H. OUERDIANEDepartment of Mathematics, Faculty of Sciences of Tunis, Université de Tunis El Manar,1060 Tunis, Tunisia. e-mail: [email protected]

(Received: 6 March 2001; in final form: 17 August 2001)

Abstract. We develop the theory of operators defined on a space of holomorphic functions. First, wecharacterize such operators by a suitable space of holomorphic functions. Next, we show that everyoperator is a limit of a sequence of convolution and multiplication operators. Finally, we define theexponential of an operator which permits us to solve some quantum stochastic differential equations.

Mathematics Subject Classifications (2000): primary 60H40; secondary 46A32, 46F25, 46G20.

Key words: symbols of operators, infinite dimensional holomorphy, convolution product of opera-tors, quantum stochastic differential equations.

1. Introduction

Let N be a complex nuclear Fréchet space. Assume that its topology is defined byan increasing family of Hilbertian norms {|.|p, p ∈ N}. Then N is represented asN =⋂

p∈NNp , where for p ∈ N the space Np is the completion of N with respect

to the norm |.|p. For simplicity, we denote by H the complex Hilbert space N0

and by N−p the dual space of Np, then the dual space N ′ of N is represented asN ′ = ⋃

p∈NN−p , and it is equipped with the inductive limit topology. We denote

by 〈., .〉 the C-bilinear form on N ′ × N connected to the inner product 〈.|.〉 of H ,i.e.

〈z, ξ 〉 = 〈z|ξ 〉, z ∈ H, ξ ∈ N.

For any n ∈ N we denote by SnN the nth symmetric tensor product of N equippedwith the π -topology and by SnNp the nth symmetric Hilbertian tensor product ofNp. We will preserve the notation |.|p and |.|−p for the norms on SnNp and SnN−p,respectively.

Let n,m ∈ N and 0 � k � m ∧ n. We denote by 〈., .〉k the bilinear map fromSmN ′ × SnN into Sm−kN ′⊗Sn−kN defined by⟨

x⊗m, y⊗n⟩k:= 〈x, y〉kx⊗(m−k) ⊗ y⊗(n−k), x ∈ N ′, y ∈ N.

The bilinear map 〈., .〉k is continuous, then using the density of the vector spacegenerated by {x⊗m, x ∈ N ′} in SmN ′ and the density of the vector space generated

66 M. BEN CHROUDA AND H. OUERDIANE

by {y⊗n, x ∈ N} in SnN , we can extend 〈., .〉k to SmN ′ ×SnN . Let φm ∈ SmN ′ andϕn ∈ SnN ; then 〈φm, ϕn〉k is called the right contraction of φm and ϕn of degree k.

Let θ be a Young function on R+, i.e. θ is continuous, convex, increasing func-tion and satisfies lim+∞ θ(x)/x = +∞. We define the conjugate function θ∗ of θ

by

∀x � 0, θ∗(x) := supt�0

(tx − θ(t)). (1)

For a such Young function θ , we denote by Gθ (N) the space of holomorphic func-tions on N with exponential growth of order θ and of arbitrary type, and by Fθ (N

′)the space of holomorphic functions on N ′ with exponential growth of order θ andof minimal type. For every p ∈ Z and m > 0, we denote by exp(Np, θ,m) thespace of entire functions f on the complex Hilbert space Np such that

nθ,p,m(f ) := supz∈Np

|f (z)|e−θ(m|z|p) < +∞.

Then the spaces Fθ (N′) and Gθ (N) are represented as

Fθ (N′) =

⋂p∈N

m>0

exp(N−p, θ,m),

Gθ (N) =⋃p∈N

m>0

exp(Np, θ,m),

and equipped with the projective limit topology and the inductive limit topology,respectively. Let p ∈ N and m > 0, we define the Hilbert spaces

Fθ,m(Np)

={f = (fn)

∞n=0, fn ∈ SnNp; ‖f ‖θ,p,m :=

∑n�0

θ−2n m−n|fn|2p < +∞

},

Gθ,m(N−p)

={φ = (φn)

∞n=0, φn ∈ SnN−p; ‖φ‖θ,−p,m :=

∑n�0

(n!θn)2mn|φn|2−p < +∞},

where θn = infr>0 eθ(r)/rn, n ∈ N. The sequences θn and θ∗n are connected by thefollowing relation

LEMMA 1. For every n ∈ N\{0} we have en � nnθnθ∗n � e2n.

Proof. We can assume that θ(x) = ∫ x

0 µ(t) dt where µ is a continuous, in-creasing function which satisfies lim+∞ µ(x) = +∞ (see [4]). Then θ∗(x) =∫ x

0 ω(t) dt , where ω is the inverse function of µ, i.e. µ ◦ ω = ω ◦ µ = id. A directcalculation shows that

θn = eθ(tn)

tnnand θ∗n =

eθ∗(xn)

xnn

,

ALGEBRAS OF OPERATORS ON HOLOMORPHIC FUNCTIONS 67

where tn and xn are the solutions of tµ(t) = n and tω(t) = n, respectively, andsatisfy tnxn = n. Hence,

nnθnθ∗n =

(n

tnxn

)n

eθ(tn)eθ∗(xn)

� e2n.

On the other hand, for every t, x > 0 we have

etx

(tx)n� eθ(t)

tn

eθ∗(x)

xn, ∀n � 1.

Then, using the fact that inft>0etx

(tx)n= en

nn , we obtain en/nn � θnθ∗n . ✷

Put

Fθ(N) =⋂p∈N

m>0

Fθ,m(Np),

Gθ (N′) =

⋃p∈N

m>0

Gθ,m(N−p).

Then the space Fθ(N) equipped with the projective limit topology is a nuclearFréchet space [4], and Gθ(N

′) carries the dual topology of Fθ(N) with respect tothe C-bilinear form (., .):

(φ,f ) =∑n�0

n!〈φn, fn〉, φ = (φn) ∈ Gθ(N′), f = (fn) ∈ Fθ(N).

For simplicity, we put

Fθ (N′) = Fθ , Gθ∗(N) = Gθ∗ , Fθ(N) = Fθ , Gθ(N

′) = Gθ

and we denote by F ′θ the strong dual of the space Fθ . It was proved in [4] that

the Taylor series map S.T yields a topological isomorphism between Fθ (respec-tively Gθ∗) and Fθ (respectively Gθ ). The nuclear Fréchet space Fθ and its dualF ′

θ are called the test function space and the distribution space, respectively. TheC-bilinear form on F ′

θ×Fθ is denoted by 〈〈., .〉〉. We denote by L(Fθ ,Fθ ) the spaceof continuous linear operators from Fθ into itself, equipped with the topology ofbounded convergence.

In this paper, we do not restrict ourselves to the theory of Gaussian (whitenoise) and non-Gaussian analysis studied, for example, in [1, 6, 8, 9] and [10]but we develop a general infinite-dimensional analysis. First, we give a decom-position of convolution operators from Fθ into itself into a sum of holomorphicderivation operators. Second, we establish a topological isomorphism between thespace L(Fθ ,Fθ ) of operators and the space Fθ ⊗Gθ∗ of holomorphic functions.


Next, we develop a new convolution calculus over L(Fθ ,Fθ ) and we give senseto the expression eT := ∑

n�0 Tn/n! for some class of operators T . Finally, as

an application of this operator theory we solve some linear quantum stochasticdifferential equations.

2. Some Properties on the Distribution Space

Let θ be a Young function. For every ξ ∈ N , the exponential function eξ : z �→e〈z,ξ 〉, z ∈ N ′ belongs to Fθ . Then we define the Laplace transform of a distributionφ ∈ F ′

θ by

φ(ξ ) := 〈〈φ, eξ 〉〉, ξ ∈ N.

PROPOSITION 1 ([4]). The Laplace transform realizes a topological isomorphismbetween F ′

θ and Gθ∗ .

By composition of the Taylor series map with the Laplace transform, we deducethat φ ∈ F ′

θ if and only if there exists a unique formal series φ = (φn)n�0 ∈ Gθ

such that

φ(ξ ) =∑n�0

〈ξ⊗n, φn〉.

Then, the action of the distribution φ on a test function ϕ(z) = ∑n�0〈z⊗n, ϕn〉 is

given by

〈〈φ, ϕ〉〉 =∑n�0

n!〈φn, ϕn〉.

In particular, for every z ∈ N ′, the Dirac mass δz defined by

〈〈δz, ϕ〉〉 = ϕ(z), (2)

belongs to F ′θ . Moreover, δz coincides with the distribution associated to the formal

series

δz :=(z⊗n

n!)

n�0

.

Now, we recall some properties of translation operators and convolution productof distributions studied in [2]. Let z ∈ N ′, the translation operator τ−z is defined by

τ−zϕ(λ) = ϕ(z+ λ), λ ∈ N ′.

For every z ∈ N ′, the linear operator τ−z is continuous from Fθ into itself. Wedefine the convolution product of a distribution φ ∈ F ′

θ with a test function ϕ ∈ Fθ

as follows

φ ∗ ϕ(z) := 〈〈φ, τ−zϕ〉〉, z ∈ N ′.


A direct calculation shows that φ ∗ ϕ ∈ Fθ . Let φ1, φ2 ∈ F ′θ , we define the

convolution product of φ1 and φ2, denoted by φ1 ∗ φ2, by

〈〈φ1 ∗ φ2, ϕ〉〉 := [φ1 ∗ (φ2 ∗ ϕ)](0), ϕ ∈ Fθ .

Moreover, ∀φ1, φ2 ∈ F ′θ we have φ1 ∗ φ2 = φ1φ2.

3. Convolution Operators

In infinite-dimensional complex analysis, a convolution operator on the test spaceFθ is a continuous linear operator from Fθ into itself which commutes with trans-lation operators. It was proved in [2, 5] that T is a convolution operator on Fθ ifand only if there exists φT ∈ F ′

θ such that

T ϕ = φT ∗ ϕ, ∀ϕ ∈ Fθ . (3)

Moreover, if the distribution φT is given by

φT = (φm)m�0 ∈ Gθ and ϕ(z) =∑n�0

〈z⊗n, ϕn〉 ∈ Fθ ,

then

φT ∗ ϕ(z) =∑m�0

∑n�0

(n+m)!n! 〈z⊗n, 〈φm, ϕm+n〉m〉. (4)

In particular, we have

T (eξ )(z) = φT ∗ eξ (z) = φ(ξ )eξ (z).

Let θ be a Young function, y ∈ N ′ and ϕ(z) = ∑n�0〈z⊗n, ϕn〉 ∈ Fθ . We define

the holomorphic derivative of ϕ at a point z ∈ N ′ in a direction y by

Dyϕ(z) :=∑n�0

(n+ 1)〈z⊗n, 〈y, ϕn+1〉1〉.

LEMMA 2. The operator Dy is continuous from Fθ into itself. Moreover, for everyϕ ∈ Fθ , p ∈ N and m > 0, we have

‖Dyϕ‖θ,p,m �√mθ1|y|−py

‖ϕ‖θ,py∨p, m16,

where py = min{p ∈ N; y ∈ N−p} and py ∨ p = max(py, p).Proof. By definition of the norm ‖.‖θ,p,m defined on the space Fθ of formal

series, we have

‖Dyϕ‖θ,p,m =( ∑

n�0

(n+ 1)2θ−2n m−n|〈y, ϕn+1〉1|2p

)1/2


� |y|−py

( ∑n�0

(n+ 1)2θ−2n m−n|ϕn+1|2p∨py

)1/2

�√m|y|−py

( ∑n�0

θ−2n+1

(m

16

)−n−1

|ϕn+1|2p∨py

[(n+ 1)θn+1

22n+2θn

]2)1/2

�√m|y|−py

supn�1

[θn+1

2n+1θn

]‖ϕ‖θ,p∨py,

m16.

Finally, the desired inequality follows immediately using the fact that 2−l−kθlθk �θl+k � 2l+kθlθk, ∀l, k ∈ N\{0}. ✷

In view of Lemma 2, for each m ∈ N the m-linear operator D defined by

D: N ′ × · · · × N ′ → L(Fθ ,Fθ )

(y1, . . . , ym) �→ Dy1 . . . Dym

is symmetric and continuous, hence, it can be continuously extended to SmN ′, i.e.D: φm ∈ SmN ′ �→ Dφm

∈ L(Fθ ,Fθ ). The action of the operator Dφmon a test

function ϕ(z) =∑n�0〈z⊗n, ϕn〉 given by

Dφm(ϕ)(z) =

∑n�0

(n+m)!n! 〈z⊗n, 〈φm, ϕn+m〉m〉. (5)

Then, in view of (3), (4) and (5), we give an expansion of convolution operatorsin terms of holomorphic derivation operators.

PROPOSITION 2. Let T ∈ L(Fθ ,Fθ ), then T is a convolution operator if andonly if there exists φ = (φm)m�0 ∈ Gθ such that T =∑

m�0 Dφm.

Remark. Let Tφ = ∑m�0 Dφm

be a convolution operator and n ∈ N. Thenequality (3) shows that

T nφ := Tφ ◦ · · · ◦ Tφ︸︷︷︸

n

= Tφ∗n . (6)

In particular,

T nφ (eξ )(z) = Tφ∗n(eξ )(z) = (φ(ξ))neξ (z), z ∈ N ′, ξ ∈ N.

4. Symbols of Operators

In this section we define the symbol map on the space L(Fθ ,Fθ ). Then we give anexpansion of such operators in terms of multiplication and derivation operators.


DEFINITION 1. Let T ∈ L(Fθ ,Fθ ), the symbol σ (T ) of the operator T is aC-valued function defined by

σ (T )(z, ξ) := e−〈z,ξ 〉T (eξ )(z), z ∈ N ′, ξ ∈ N.

Similar definitions of symbols have been introduced in various contexts ([7, 10–12]).

In the general theory ([13]), if we take two nuclear Fréchet spaces X and D ,then the canonical correspondence T ↔ KT given by

〈T u, v〉 = 〈KT , u⊗ v〉, u ∈ X, v ∈ D ′,

yields a topological isomorphism between the spaces L(X,D) and X′ ⊗D . Inparticular, if we take X = D = Fθ which is a nuclear Fréchet space, then we get

L(Fθ ,Fθ ) ∼= F ′θ ⊗ Fθ . (7)

So, the symbol σ (T ) of an operator T can be regarded as the Laplace transform ofthe kernel KT

σ(T )(z, ξ) = KT (eξ ⊗ δz), z ∈ N ′, ξ ∈ N. (8)

Moreover, with the help of equalities (2), (7), (8) and Proposition 1 we obtain thefollowing theorem

THEOREM 1. The symbol map yields a topological isomorphism betweenL(Fθ ,Fθ ) and Fθ⊗Gθ∗ . More precisely, we have the following isomorphisms:

L(Fθ ,Fθ )σ→ Fθ⊗Gθ∗

S.T→ Fθ⊗Gθ,

T �→ σ (T )(z, ξ) =∑l,m

〈Kl,m, z⊗l ⊗ ξ⊗m〉 �→ K = (Kl,m)l,m�0.

EXAMPLES. (1) Let φm ∈ SmN ′. Then

σ (Dφm)(z, ξ) = e−〈z,ξ 〉Dφm

(eξ )(z)

= e−〈z,ξ 〉〈φm, ξ⊗m〉e〈z,ξ 〉

= 〈φm, ξ⊗m〉.

In particular, the symbol of a convolution operator Tφ =∑m�0 Dφm

is given by

σ (Tφ)(z, ξ) = e−〈z,ξ 〉∑m�0

Dφm(eξ )(z) =

∑m�0

〈φm, ξ⊗m〉 = φ(ξ ).

Hence, the operator Tφ can be expressed in an obvious way by

Tφ =∑m�0

Dφm:=

∑m�0

〈φm,D⊗m〉 = σ (Tφ)(z,D), z ∈ N ′.


(2) Let f ∈ Fθ . We denote by Mf the multiplication operator by the testfunction f . Its symbol is given by

σ (Mf )(z, ξ) = e−〈z,ξ 〉(f eξ )(z)

= e−〈z,ξ 〉f (z)eξ (z)

= f (z).

By the same argument, the multiplication operator is also expressed by Mf =σ (Mf )(z,D). We note that the symbol of a convolution (respectively, multiplica-tion) operator σ (T )(z, ξ) depends only on ξ (respectively, z).

Let K ∈ Fθ⊗Gθ and assume that K = f ⊗ φ = (fl ⊗ φm)l,m�0. Then theoperator T associated to K (see Theorem 1) satisfies

T = MfTφ, (9)

where f (z) = ∑l�0〈z⊗l , fl〉 and Tφ is the convolution operator associated to the

distribution φ given by φ. Moreover, we have

T = MfTφ = σ (Mf )(z,D)σ (Tφ)(z,D) = σ (T )(z,D).

Thus, using the density of Fθ ⊗Gθ in Fθ ⊗Gθ , we obtain the following result:

PROPOSITION 3. The vector space generated by operators of type (9) is densein L(Fθ ,Fθ ).

5. Convolution Product of Operators

Let T1, T2 be two operators in L(Fθ ,Fθ ); the convolution product of T1 and T2,denoted by T1 ∗ T2, is uniquely determined by

σ (T1 ∗ T2) = σ (T1)σ (T2).

If the operators T1 and T2 are of type (9), i.e. T1 = Mf1Tφ1 and T2 = Mf2Tφ2 , then

T1 ∗ T2 = Mf1f2Tφ1∗φ2 .

In particular, if T = MfTφ, then for every n ∈ N we have

T ∗n = MfnTφ∗n . (10)

Remark. Let Tφ (resp. Mf ) be a convolution (resp. multiplication) operator.Then for every n ∈ N

T ∗nφ = Tφ∗n = T nφ and M∗n

f = Mfn = Mnf .


LEMMA 3. Let γ1, γ2 two Young functions and (Fn) a sequence belonging toFγ1⊗Gγ2 . Then (Fn) converges in Fγ1⊗Gγ2 if and only if

(1) (Fn) is bounded in Fγ1⊗Gγ2 .(2) (Fn) converges simply.

Proof. The proof is based on the nuclearity of the spaces Fγ1 and Gγ2 . A similarproof is established with more details in [3], Theorem 2. ✷PROPOSITION 4. Let T ∈ L(Fθ ,Fθ ); then the operator

e∗T :=∑n�0

T ∗n

n!belongs to L(F(eθ∗)∗,Feθ ).

Proof. Let T ∈ L(Fθ ,Fθ ) and put

Sn =n∑

k=0

T ∗k

k! .

Then, using Lemma 3, we show that σ (Sn) converges in Feθ ⊗Geθ to eσ(T ), fromwhich the assertion follows. ✷COROLLARY 1. Let T ∈ L(Fθ ,Fθ ), and assume that σ (T )(z, ξ) is a polyno-mial in z and ξ of degree k and k/(k− 1), respectively, k � 2. Then e∗T belongs toL(Fk,Fk), where Fk is the test space associated to the Young function θ(x) = xk .

Let T ∈ L(Fθ ,Fθ ) and consider the linear differential equation

dE

dt= TE, E(0) = I.

Then the solution is given informally by

E(t) = etT , t ∈ R.

In the particular case, where T is a convolution or a multiplication operator; thesolution E(t) = etT is well defined since eT = e∗T . If T is not a convolution or amultiplication operator then the following theorem gives a sufficient condition onT to insure the existence of its exponential eT .

THEOREM 2. Let K = (Kl,m) ∈ Fθ⊗Gθ satisfying 〈Kl,m,Kl′,m′ 〉k = 0 for everym, l′ � 1,m′, l � 0 and 1 � k � m ∧ l′ and denote by T the operator associatedto K (see Theorem 1). Then,

T n = T ∗n, ∀n ∈ N.

Moreover, eT = e∗T ∈ L(F(eθ∗)∗,Feθ ).


Proof. Using Proposition 3, it will be sufficient to assume that Kl,m = (fl⊗φm),i.e.

T = MfTφ =∑l,m�0

MflDφm

,

where fl(z) = 〈z⊗l , fl〉. Assume that

fl = η⊗l , η ∈ N and φm = y⊗m, y ∈ N ′.

Then it is easy to see that

DφmMfl

= MflDφm

+m∧l∑k=0

k!Ckl C

km〈y, η〉kMfl−k

Dφm−k,

an equality on Fθ . The assumption 〈Kl,m,Kl′,m′ 〉k = 0 implies that 〈y, η〉 = 0.Then

DφmMfl

= MflDφm

. (11)

Thus, using the density of the vector space generated by {η⊗l , η ∈ N} in the spaceSlN and the density of the vector space generated by {y⊗m, y ∈ N ′} in SmN ′, wecan extend equality (11) to every fl ∈ SlN and φm ∈ SmN ′ such that 〈φm, fl〉k =0,∀1 � k � l ∧m. Hence, we obtain

MfTφ =∑l,m�0

MflDφm

=∑l,m�0

DφmMfl

= TφMf .

Using equalities (6) and (10), for every n ∈ N we have

T n = (Mf Tφ)n = (Mf )

n(Tφ)n = MfnTφ∗n = T ∗n.

This completes the proof. ✷Remark. The condition of Theorem 2 is not satisfied by convolution or multi-

plication operators. In fact, let K = (Kl,m) ∈ Fθ⊗Gθ and let T be the operatorassociated to K .

If T is a convolution operator then K = (Kl,m)l,m�0 = (K0,m)m�0 ∈ Gθ , seeProposition 2. Hence, the right contraction 〈Kl,m,Kl′,m′ 〉k = 0 with 1 � k � m∧ l′can not be defined since l′ = 0.

If T is a multiplication operator then K = (Kl,m)l,m�0 = (Kl,0)l�0 ∈ Fθ . Thus〈Kl,m,Kl′,m′ 〉k = 0 with 1 � k � m ∧ l′ can not be defined since m = 0.

Now we give an example of family of kernels K ∈ Fθ⊗Gθ which satisfies thecondition of Theorem 2.

EXAMPLE. Let N = S(R) ↪→ H = L2(R, dt) ↪→ N ′ = S ′(R) and K =(Kl,m)l,m�0 ∈ Fθ ⊗Gθ , i.e. Kl,m ∈ Sl(S(R))⊗Sm(S ′(R)). Assume that there exists


t ∈ R such that for every l, m ∈ N the support of Kl,m is included in ]−∞, t]l×]t ,+∞[m. Then K satisfies the condition of Theorem 2.

Remark. In Theorem 2 we assume that N is a C-vector space of dimensionn � 2. However, if N = C then for every m, l � 0; SlN⊗SmN ′ = C. Thus theassumption 〈Kl,m,Kl′,m′ 〉k = 0 for every m, l′ � 1, m′, l � 0 and 1 � k � m∧l′ isequivalent to Kl,m = 0, ∀l, m ∈ N and the set of operators satisfying the conditionof Theorem 2 is reduced to the null operator.

6. Applications to Quantum Stochastic Differential Equations

A one-parameter quantum stochastic process with values in L(Fθ ,Fθ ) is a familyof operators {Et, t ∈ [0, T ]} ⊂ L(Fθ ,Fθ ) such that the map t �→ Et is continuous.For a such quantum process Et we set

En = t

n

n−1∑k=0

Etkn, n ∈ N\{0}, t ∈ [0, T ].

Then we prove using Lemma 3 that the sequence (En) converge in L(Fθ ,Fθ ). Wedenote its limit by∫ t

0Es ds := lim

n→+∞En in L(Fθ ,Fθ ).

Moreover, we have

σ

(∫ t

0Es ds

)=

∫ t

0σ (Es) ds, ∀t ∈ [0, T ].

THEOREM 3. Let t ∈ [0, T ] �→ f (t) ∈ Fθ and t ∈ [0, T ] �→ φ(t) ∈ F ′θ be

two continuous processes and put Lt = Mf(t)Tφ(t). Then the linear differentialequation

dEt

dt= Mf(t)EtTφ(t), E0 = I (12)

has a unique solution Et ∈ L(F(eθ∗)∗,Feθ ) given by

Et = e∗(∫ t

0 Ls ds).

Proof. Applying the symbol map to Equation (12) we get

dσ (Et )

dt= σ (Lt)σ (Et), σ (I ) = 1.

Then σ (Et) = e∫ t

0 σ(Ls) ds which is equivalent to Et = e∗(∫ t

0 Ls ds). Finally, weconclude by Proposition 4 that Et ∈ L(F(eθ∗)∗,Feθ ). ✷


THEOREM 4. Let Lt be a quantum stochastic process with values in L(Fθ ,Fθ )

such that

σ

(∫ t

0Ls ds

)(z, ξ) =

∑l,m�0

〈Kl,m(t), z⊗l ⊗ η⊗m〉,

and assume that for every t ∈ [0, T ],m′, l � 0 and m, l′ � 1 we have

〈Kl,m(t),Kl′,m′(t)〉k = 0, ∀1 � k � m ∧ l′.

Then the following differential equation

dE

dt= LtE, E(0) = I, (13)

has a unique solution in L(F(eθ∗)∗,Feθ ) given by E(t) = e∫ t

0 Ls ds .

Acknowledgement

We are grateful to the Professor Luis Boutet de Monvel for many stimulatingremarks and useful suggestions.

References

1. Albeverio, S., Daletsky, Yu. L., Kondratiev, Yu. G. and Streit, L.: Non-Gaussian infinitedimensional analysis, J. Funct. Anal. 138 (1996), 311–350.

2. Ben Chrouda, M., Eloued, M. and Ouerdiane, H.: Convolution calculus and applications tostochastic differential equations, To appear in Soochow J. Math. (2001).

3. Ben Chrouda, M., Eloued, M. and Ouerdiane, H.: Quantum stochastic processes and applica-tions, Preprint, 2001.

4. Gannoun, R., Hachaichi, R., Ouerdiane, H. and Rezgui, A.: Un théorème de dualité entre espacede fonctions holomorphes à croissance exponentielle, J. Funct. Anal. 171(1) (2000), 1–14.

5. Gannoun, R., Hachaichi, R., Krée, P. and Ouerdiane, H.: Division de fonction holomorphe acroissance θ-exponentielle, Preprint, BiBos No. E 00-01-04, 2000.

6. Hida, T., Kuo, H.-H., Potthof, J. and Streit, L.: White Noise, An Infinite-Dimentional Calculus,Kluwer Acad. Publ., Dordrecht, 1993.

7. Krée, P. and Raczka, R.: Kernels and symbols of operators in quantum field theory, Ann. Inst.H. Poincaré Sect. A 18(1) (1978), 41–73.

8. Kondratiev, Yu. G., Streit, L., Westerkamp, W. and Yan, J.-A.: Generalized functions in infinitedimensional analysis, Hiroshima Math. J. 28 (1998), 213–260.

9. Kuo, H.-H.: White Noise Distribution Theory, CRC Press, Boca Raton, 1996.10. Obata, N.: White Noise Calculus and Fock Space, Lecture Notes in Math. 1577, Springer, New

York, 1994.11. Obata, N.: Wick product of white noise operators and quantum stochastic differential equations,

J. Math. Soc. Japan 51(3) (1999), 613–641.12. Ouerdiane, H.: Noyaux et symboles d’opérateurs sur des fonctionnelles analytiques gaussi-

ennes, Japan. J. Math. 21(1) (1995), 223–234.13. Trèves, F.: Topological Vector Space, Distributions and Kernels, Academic Press, New York,

1967.


77

On the Gaussian Perceptron at High Temperature

MICHEL TALAGRANDEquipe d’Analyse-Tour 46, ESA au CNRS No. 7064, Université Paris VI, 4 Pl. Jussieu,75230 Paris Cedex 05, France, and Department of Mathematics, The Ohio State University,231 W. 18th Ave., Columbus, OH 43210-1174, U.S.A.

(Received: 9 April 2001; in final form: 25 September 2001)

Abstract. For σ = (σi)i�N ∈ �N = {−1, 1}N , define

H(σ ) = −∑k�M

u

(1√N

∑i�N

σigki

),

where (gki )i�N,k�M are i.i.d. N(0, 1), and where u is bounded and Borel measurable. When M is asmall proportion α of N , we study the system with random Hamiltonian H , at temperature 1. Whenα is small enough, we prove that the overlap of two configurations taken independently at randomfor Gibbs’ measure is nearly constant, with a correct estimate of the size of its fluctuations.

Mathematics Subject Classifications (2000): Primary: 82D30; secondary: 60D05.

Key words: replica-symmetry, pure state, perceptron.

1. Introduction

Physicists have developed a remarkable theory of mean field disordered systems[MPV], but the study of these is still in its infancy. The physicists rely upon anumber of intuitions or, if one prefers, of heuristic general principles. One of theseprinciples is that ‘at high temperature, the overlap of two configurations chosenindependently at random according to Gibbs’ measure is nearly constant’. (Theoverlap of two configurations is defined after the statement of Theorem 1.1.) Eventhough this principle emerged from physical experience, the physicists have ap-parently no qualms to apply it to mathematical objects (such as the one that willbe considered here) that are certainly not realistic models for interactions withinactual matter. This bold approach seems to yield correct results. This was recentlyrigorously proved for four of the most popular models (see [T1] for a survey). Thecase that will be considered here offers a new difficulty (a type of discontinuity).This difficulty appeared serious enough at first sight to have the author doubt thatthe result should be true. To these doubts, M. Mézard gave in essence the followingvery interesting answer: ‘But if the system is not in a pure state, what else?’ Whatelse indeed, and whether the physical principles can be supported by a general

78 MICHEL TALAGRAND

mathematical principle (rather than by difficult proofs in each special case) arefood for further thought.

Consider a bounded function u:R → R. Consider i.i.d. N(0, 1) r.v. (gki )i�N,k�M ,

that represent the ‘disorder’ of the system, and consider the random Hamiltonian

HN,M(σ ) = −∑k�M

u

(1√N

∑i�N

gki σi

). (1.1)

We are interested in the behavior of the system governed by the Hamiltonian (1.1)at inverse temperature 1, that is in the (random Gibbs’) probability measure GN,M

on �N defined by

GN,M(σ ) = Z−1N,M exp(−HN,M(σ )), (1.2)

where ZN,M is the normalization factor

ZN,M =∑

σ

exp(−HN,M(σ )).

(The reason for which we do not consider the usual inverse temperature parameterβ is that it can be included in u.) The reason for the name ‘perceptron’ is that if

u = β1{x�τ }, (1.3)

then (hopefully) as β → ∞ the knowledge of GN,M allows to recover informationon

card

{σ ; ∀k � M,

1√N

∑i�N

gki σi � τ

}, (1.4)

a problem referred to in the neural networks theory as ‘The problem of the capacityof the Gaussian perceptron’. The reason for the term ‘Gaussian’ is that the ran-dom variables (gk

i ) are i.i.d. N(0, 1), while in the usual perceptron they are ratherBernoulli (P (gk

i = ±1) = 1/2). The choice of Gaussian r.v. is more natural fromthe point of view of geometry since then in (1.3) the sets {σ ;∑i�N gk

i σi � τ√N}

are random half-spaces at (nearly) fixed distance from the origin, with a uniformlyrandom direction. The reason why we consider the Gaussian case is simply that thisis easier than the Bernoulli case (the results of the present paper are probably truein the Bernoulli case, but we doubt that they are within reach of todays methods).

The theory of neural networks is a rich theory full of promises. The problem ofthe capacity of the perceptron is of fundamental importance in this theory. For thepresent purpose however, only the geometric formulation we gave is relevant, sowe send the reader to [HKP] for a general and readable introduction about neuralnetworks. We also refer the reader to [GD] for the (nonrigorous) approach by thephysicists.

We will always assume that M is a fixed proportion of N,M = αN�, whereα > 0.

ON THE GAUSSIAN PERCEPTRON AT HIGH TEMPERATURE 79

In a previous paper [T2], we studied the present problem in the Bernoulli case(but the Gaussian case should be similar) under the additional hypothesis that thefirst five derivatives of u exist and are bounded, say

� � 5 ⇒ |u(�)| � D′. (1.5)

Then we reached a good understanding of the system provided

N � N(D′), (1.6)

Lα expLD � 1, (1.7)

where D = sup|u| and where L is a number. It is important that the condition(1.7) does not depend upon D′. Unfortunately this result is not useful when u isnot five times differentiable and in particular in the case (1.3) (although we couldcalculate the important quantity limN→∞ N−1E log ZN,M by approximating u bya differentiable function). The purpose of the present paper is twofold. First, wewant to remove conditions (1.5), (1.6), and assume only minimum regularity foru, so as to cover in particular the case (1.3) when β is small enough. Second,we want to obtain the correct rates of convergence, rates that cannot be reachedwith the previous methods. The reader might think at first that removing a meresmoothness condition is not a big deal, but this has actually required considerableeffort, and as a result of these efforts, the methods are not more complicated, butare considerably more powerful than those of [T2]. Moreover, these methods haveyielded considerable simplifications of the results previously obtained for othermodels, as is demonstrated in [T3] for the Sherrington–Kirkpatrick model.

Let us explain the basic difficulty. The natural approach is the cavity method,that relates an (N + 1) spin system with an N-spin system. The expert certainlyguesses that when attempting this, one meets the quantity

� =∑k�M

(u(Sk

√N/N ′ + gk

√1/N ′)− u(Sk)

), (1.8)

where

Sk = N−1/2∑i�N

gki σi, N ′ = N + 1

and (gk)k�M is a sequence of N(0, 1) r.v. independent of the (gki ). (The nonexpert

will of course be explained in detail the variation of the cavity method we willneed, for which the difficulties are of the same nature.) When u is three timesdifferentiable, we can write, by a simple computation, that (1.8) is

� �∑k�M

(gk√N ′u

′(Sk)+ 1

2N ′ (g2ku

′′(Sk)− Sku′(Sk))

)(1.9)

within an error R, where |R| � ‖u(3)‖∞/√N . Thus, as N → ∞, R vanishes,

leaving us with a manageable expression for (1.8). On the other hand, if ‖u(3)‖∞

80 MICHEL TALAGRAND

does not remain bounded with N , there seems to be no reasonable way to express(1.8), and no reason why it should depend upon u′, u′′ only. Yet, for some purposesthis is the case. Indeed, if we denote by 〈·〉 averages for Gibbs’ measure, we knowhow to prove that

〈exp �〉 � exp∑k�M

[gk√N ′ 〈u

′(Sk)〉 +

+ 1

2N(〈u′′(Sk)+ u′2(Sk)− Sku

′(Sk)〉 − 〈u′(Sk)〉2)

], (1.10)

where � means that the error is typically at most L/√N . To appreciate this for-

mula, we observe that by (1.9), � is of order 1 for large N when ‖u(3)‖∞ < ∞.On the other hand, in the case (1.3) (or of the very small perturbation we willconsider) � is the sum of M terms, each of which having a chance of order 1/

√N

to be ±1, so one expects that 〈�2〉 is typically of order α√N . But the right-hand

side of (1.10) is order 1; so that in order for (1.10) to hold, rather extraordinarycancelation has to take place. Extraordinary cancelation is indeed the theme of thepaper.

We now state precisely our results. Throughout the paper, L denotes a number(independent of everything) that need not be the same at each occurrence.

We assume throughout the paper that

|u| � D. (1.11)

We consider two N(0, 1) variables h, z, and for x ∈ R, 0 < y � 1,

"(x, y) = Eh exp u(x + hy)

yE exp u(x + hy). (1.12)

We consider the systems of equations

q = Eth2(z√q), (1.13)

q = αE"2(z√q,√

1 − q). (1.14)

We leave the reader to check (see [T2] for more details) that if (for a number L

large enough)

Lα expLD � 1, (1.15)

then there is a unique solution (q, q) to (1.13), (1.14). Of course, the meaning ofthese equations is not so obvious, and the fact that this precise value of q appearsnaturally is, in a sense, the proof that we are dealing with a subtle and rich situation.

THEOREM 1.1. If u satisfies (1.11) and is Borel measurable, then we have under(1.15) that

E

⟨(σ · σ ′

N− q

)2⟩� L

N. (1.16)


In (1.16), σ · σ ′ = ∑i�N σiσ

′i for σ , σ ′ in �N , and the bracket 〈·〉 represents

a double integral on �2N with respect to G⊗2

N,M . The quantity σ · σ ′/N is calledthe overlap of the two configurations σ , σ ′, and Theorem 1.1 expresses that theoverlap of two configurations chosen independently at random according to Gibbs’measure is nearly q. The meaning of this condition is not so intuitive either, and werefer the reader to [T3] for a detailed explanation of its fundamental importance inthe case of another model, the Sherington–Kirkpatrick model.

We will deduce Theorem 1.1 from the following theorem:

THEOREM 1.2. There exists a number L with the following property. If u satisfies(1.11), is ten times differentiable, and satisfies

∀� � 10, |u(�)| � exp(N/L), (1.17)

then (1.16) holds under (1.15).

Let us now comment briefly upon the methods of the paper. Our answer tothe problem of how to evaluate quantities such as (1.8) is that we do not try todo this. Rather, we use the idea to move along a suitably chosen continuous pathfrom a simple situation to the situation we want to study (Kahane’s principle).The derivatives along the path are studied using integration by parts, on which allthe cancelations explained above ultimately rely. In [T2], the cavity argument wasbroken into a ‘cavity upon N’ part and a ‘cavity upon M part’. It does not seem tobe possible to do this here, and both parts of the arguments are combined. Rather,Kahane’s principle has to be used twice. The ‘bottom’ use is the object of Section 2.It is a more powerful version of Lemma 3.2 of [T2]. The ‘top’ use, a kind of cavityargument, is the object of Section 3, that culminates in the proof of Theorem 1.2.The short, final section deduces Theorem 1.1 from Theorem 1.2.

2. Integration by Parts

The basic integration by parts principle we will use is that if f is a smooth functionof moderate growth, and if g is a centered Gaussian r.v., then

E(gf (g)) = Eg2Ef ′(g). (2.1)

Here is a simple consequence.

LEMMA 2.1. Consider a centered Gaussian family g1, . . . , gm. We assume thatfor each � � m, we can write

g� = gk,� +∑k �=�

bk,�gk, (2.2)

where gk,� is independent of g1, . . . , g�−1, g�+1, . . . , gk and where

Eg2k,� � 1

4;

∑k �=�

|bk,�| � 1. (2.3)

82 MICHEL TALAGRAND

Consider a smooth function F on Rm. Then for each integers r1, . . . , rm, s1, . . . , sm

we have∣∣∣∣E(gr11 . . . grm

m

∂s

∂xs11 . . . ∂x

smm

F(g1, . . . , gm)

)∣∣∣∣ � K sup|F |, (2.4)

where s = s1 + · · · + sm and where K depends only upon s1, . . . , sm, r1, . . . , rm.Proof. The proof goes by induction upon s. Certainly the result is true for s = 0.

For the induction step, we can assume s1 � 1, and using (2.2) we can write g1 =g +∑

k�2 bkgk, where∑

k�1 |bk| � 1, g is independent of g2, . . . , gm and Eg2 �1/4. We then observe that by (2.1), for any number a, we have

E(g(g + a)r1f (g + a))

= Eg2(r1E((g + a)r1−1f (g + a))+ E((g + a)r1f ′(g + a)). (2.5)

Using this for a =∑k�2 bkgk at g2, . . . , gm fixed, for

f (x) = ∂s−1

∂xs1−11 ∂x

s22 . . . ∂x

smm

F(x, g2, . . . , gm),

we obtain

E(gr11 . . . grm

m

∂s

∂xs11 ∂x

s22 . . . ∂x

smm

F(g1, . . . , gm))

= 1

Eg2E

((g1 −

∑k�2

bkgk)gr11 . . . grm

m

∂s−1

∂xs1−11 ∂x

s22 . . . ∂x

smm

F(g1, . . . , gm)

)−

− r1E

(gr1−11 g

r22 . . . grm

m

∂s−1

∂xs1−11 ∂x

s2s . . . ∂x

smm

F(g1, . . . , gm)

)

and this implies the result. ✷We now present a simple condition that ensures the conditions (2.2), (2.3) of

Lemma 2.1.

LEMMA 2.2. Assume that the Gaussian r.v. (g�)��m satisfy

∀� � m,Eg2� � 3

4; ∀� < �′ � m, |Eg�g�′ | � 1

4m. (2.6)

Then the conditions of Lemma 1.2 hold.Proof. To prove (2.2) we assume without loss of generality that � = 1, and we

write

g1 = g +∑k�2

bkgk, (2.7)


where g is independent of g2, . . . , gm. If j � 2, we deduce from (2.7) that

bjEg2j = Eg1gj −

∑k�2,k �=j

bkEgkgj ,

so that

34 |bj | � 1

4m

(1 +

∑k�2

|bk|)

and by summation over j � 2,

34

∑j�2

|bj | � 14

(1 +

∑k�2

|bk|),

so that∑j�2

|bj | � 12 . (2.8)

Now, from (2.7) again, we get

Eg21 = Egg1 +

∑k�2

bkEg1gk

� Egg1 + 1

4m

∑k�2

|bk| � Egg1 + 1

8m,

so that

Egg1 � Eg21 −

1

8m� 5

6Eg21

and, since Egg1 � (Eg2)1/2(Eg21)

1/2, we get

Eg2 �(

56

)2Eg2

1 � 34

(56

)2 � 14 . ✷

We consider now a finite set J , and a probability measure µ on J . We consider amap f from Jm to [−1, 1], and two functions U,V on R

m. We consider a centeredGaussian family (g(j))j∈J such that

∀j, Eg(j)2 � 34 . (2.9)

We consider the quantity

I = E

∫f (j1, . . . , jm)U(g(j1), . . . , g(jm)) dµ(j1) . . . dµ(jm)∫

V (g(j1), . . . , g(jm)) dµ(j1) . . . dµ(jm). (2.10)

84 MICHEL TALAGRAND

LEMMA 2.3. We consider numbers B, C, C ′ > 0. We assume that

V � 1

B, (2.11)

|U | � C ′, (2.12)

U(x1, . . . , xm) = ∂s

∂xs11 . . . ∂x

smm

F(x1, . . . , xm), (2.13)

where

|F | � C. (2.14)

We set

A = µ⊗({

(j1, j2); |Eg(j1)g(j2)| � 1

8m

}). (2.15)

Then we have

|I | � KB

[C

∫|f (j1, . . . , jm)| dµ(j1) . . . dµ(jm)+ C ′A1/2

], (2.16)

where K depends only upon m, s1, . . . sm.

Comment. This will be used in situations where C ′ � C, but where A isextremely small, so the term C ′A1/2 will be very small.

Proof. We use the Cauchy–Schwarz inequality to write, with obvious short-handnotation,

I �(E(∫ fU)2

(∫V )2

)1/2

� B

(E

(∫fU

)2)1/2

, (2.17)

using (2.11). Now

E

(∫fU

)2

�∫

f (j1, . . . , jm)f (jm+1, . . . , j2m)×× E(U(g(j1), . . . , g(jm))×× U(g(jm+1), . . . , g(j2m))) dµ(j1) . . . dµ(j2m). (2.18)

We write∫=∫61

+∫6c

1

, (2.19)

where

61 ={j1, . . . , j2m, ∃� < �′ � 2m E|g(j�)g(j�′)| � 1

8m

}.


We observe that

µ⊗2m(61) � m(2m− 1)A, (2.20)

so that, using (2.12),∣∣∣∣∫61

∣∣∣∣ � m(2m− 1)AC2.

On 6c1, we use Lemmas 2.1 and 2.2 (with 2m rather than m) to bound the integrand

by

KC2|f (j1, . . . , jm)| |f (jm+1, . . . , j2m)| (2.21)

(where K depends only upon m, s1, . . . , sm) so that∣∣∣∣∫6c

1

∣∣∣∣ � KC2

(∫|f (j1, . . . , jm)| dµ(j1) . . . dµ(jm)

)2

. ✷

3. The Main Estimate

In this section, we show how to approximate the quantity (2.10). We consider anumber 0 � q � 1/24m and independent N(0, 1) r.v. z, h1, . . . hm. We denote byEh expectation at z given, i.e. in h1, . . . , hm only.

THEOREM 3.1. Assume the conditions of Lemma 2.3, and that, moreover,

|V | � C; ∀�, �′ � m,

∣∣∣∣ ∂V∂x�∣∣∣∣ � C ′;∣∣∣∣ ∂2V

∂x�∂x�

∣∣∣∣ � C ′,∣∣∣∣∂U∂x�

∣∣∣∣ � C ′;∣∣∣∣ ∂2U

∂x�∂x�′

∣∣∣∣ � C ′. (3.1)

Then, for some constant K depending only upon m, s1, . . . , sm, we have∣∣∣∣I −∫

f (j1, . . . , jm)dµ(j1) . . . dµ(jm)×

×EEhU(z

√g + h1

√1 − g, . . . , z

√g + hm

√1 − g)

EhV (z√g + h1

√1 − g, . . . , z

√g + hm

√1 − g)

∣∣∣∣� KB3C3

(∫f 2(j1, . . . , jm) dµ(j1) . . . dµ(jm)

)1/2

×

×(∫

(Eg(j1)g(j2)− q)2 dµ(j1) dµ(j2)

)1/2

+

+(∫

(Eg(j)2 − 1)2 dµ(j)

)1/2]+KB3C ′3A1/2, (3.2)

86 MICHEL TALAGRAND

where

A = µ⊗2

{(j, j ′); |Eg(j)g(j ′)| > 1

24m

}. (3.3)

Comment. To make this result useful, we will of course choose q to make theright-hand side small.

Proof. It will be helpful to consider first the case where µ is replaced by

µ′ = 1

R

∑r�R

δjr , (3.4)

where the points j1, . . . , jR of J need not be distinct. We set J ′ = {1, . . . , R},we denote by γ the uniform probability on J ′. To lighten notation, we write g(r)

rather than g(jr) and f (r1, . . . , rm) rather than f (jr1, . . . , jrm). We consider

I ′ = E

∫f (r1, . . . , rm)U(g(r1), . . . , g(rm)) dγ (r1) . . . dγ (rm)∫

V (g(r1), . . . , g(rm)) dγ (r1) . . . dγ (rm). (3.5)

This corresponds to the quantity (2.10) when µ has been replaced by µ′.We consider i.i.d. N(0, 1) r.v. h(r), r � R, and

ξ(r) = z√q + h(r)

√1 − q.

We note that

Eξ(r)ξ(r ′) ={q if r �= r ′,1 if r = r ′.

(3.6)

For 0 � t � 1 we consider

ξt (r) =√t g(r)+√

1 − t ξ(r).

We consider the function

ψ(t) = E

∫f (r1, . . . , rm)U(ξt (r1), . . . , ξt (rm)) dγ (r1) . . . dγ (rm)∫

V (ξt(r1), . . . , ξt (rm)) dγ (r1) . . . dγ (rm). (3.7)

Thus ψ(1) = I ′. We write

|I ′ − ψ(0)| = |ψ(1)− ψ(0)| �∫ 1

0|ψ ′(t)| dt � sup

0<t<1|ψ ′(t)|. (3.8)

This formula bounds the error we make when we approximate I ′ by the simplerquantity ψ(0). (The reason why ψ(0) is simpler than I is that we have replacedthe family (g(r)) by the simpler family (ξ(r)).) The rest of the proof consists ofestimating the right-hand side of (3.8). Then we will make R !→ ∞ and µ′ !→ µ

to obtain (3.2).


To compute ψ ′, we observe that

ξ ′t (r) :=d

dtξt (r) = 1

2

(1√tg(r)− 1√

1 − tξ(r)

), (3.9)

so that

Eξ ′t (r)ξt (r′) = 1

2(Eg(r)g(r ′)− Eξ(r)ξ(r ′))

:= �(r, r ′). (3.10)

With obvious notation, we have

ψ ′(t) =∑��m

E

(1

D

∫f∂U

∂x�ξ ′t (r�)

)−

−∑��m

E

(1

D2

∫fU

∫∂V

∂x�ξ ′t (r�)

)(3.11)

for

D =∫

V =∫

V (ξt(r1), . . . , ξt (rm)) dγ (r1) . . . dγ (rm). (3.12)

To make sense of (3.11), we integrate by parts, using (3.10), and the followingeasy generalization of (2.1):

E(gf (g1, . . . , gm)) =∑��m

E(gg�)E∂f

∂x�(g1, . . . , gm). (3.13)

This integration by parts is a cumbersome but straightforward computation, theresult of which is that

ψ ′(t) = I + II + III + IV, (3.14)

where

I =∑

�,�′�m

E

(1

D

∫f

∂2U

∂x�∂x�′(ξt (r1), . . . , ξt (rm))×

×�(r�, r�′) dγ (r1) . . . dγ (rm)

), (3.15)

II = −2∑

�,�′�m

E

(1

D2

∫f∂U

∂x�(ξt (r1), . . . , ξt (rm))×

× ∂V

∂x�′(ξt (rm+1), . . . , ξt (r2m))�(r�, rm+�′) dγ (r1) . . . dγ (r2m)

), (3.16)

88 MICHEL TALAGRAND

III = −∑

�,�′�m

E

(1

D2

∫fU(ξt(r1), . . . , ξt (rm))×

× ∂2V

∂x�∂x�′(ξt (rm+1), . . . , ξt (r2m))×

×�(rm+�, rm+�′) dγ (r1) . . . dγ (r2m)

), (3.17)

IV = 2∑

�,�′�m

E

(1

D3

∫fU(ξt(r1), . . . , ξt (rm))×

× ∂V

∂x�(ξt (rm+1), . . . , ξt (r2m))

∂V

∂x�′(ξt (r2m+1), . . . , ξt (r3m))×

×�(rm+�, r2m+�′) dγ (r1) . . . dγ (r3m)

). (3.18)

In these formulas, f = f (r1, . . . , rm). Even though these formulas look com-plicated, the good news is that all these terms are of the same nature as (2.10),provided we replace m by m′ = 2m or by m′ = 3m. We will be able to use thebound (2.16), replacing B by B3, C by C3, C ′ by C ′3, and the quantity A of (2.15)by

γ ⊗2

({(r, r ′); |Eξt(r)ξt (r

′)| � 1

24m

}). (3.19)

Since for r �= r ′,

ξt (r)ξt (r′) = tE(g(r)g(r ′))+ (1 − t)q

and since we assume q � 1/24m, the quantity (3.19) is bounded by

A′ = γ ⊗2

({(r, r ′); |E(g(r)g(r ′))| � 1

24m

}).

We also observe that if �, �′ � 3m, then∫|f (r1, . . . , rm)�(r�, r�′)| dγ (r1) . . . dγ (r3m)

�(∫

f 2(r1, . . . , rm) dγ (r1) . . . dγ (rm)

)1/2

×

×(∫

�2(r�, r�′) dγ (r�) dγ (r�′)

)1/2

. (3.20)

Thus, (2.16), (3.14), (3.17) imply that for a constant K depending only on m,s1, . . . , sm, we have

|ψ ′(t)| � KB3C3

(∫f 2(r1, . . . , rm) dγ (r1) . . . dγ (rm)

)1/2

×


×[( ∫

�2(r1, r2) dγ (r1) dγ (r2)

)1/2

+

+(∫

�2(r, r) dγ (r)

)1/2]+KB3C ′ 3A′ 1/2. (3.21)

As µ′ converges to µ, the right-hand side of (3.21) converges to the right-hand sideof (3.2), and I ′ converges to I . Thus, it remains to study how ψ(0) behaves. Wehave ψ(0) = E(W/S), where

W =∫

f (r1, . . . , rm)U(ξ(r1), . . . , ξ(rm)) dγ (r1) . . . dγ (rm), (3.22)

S =∫

V (ξ(r1), . . . , ξ(rm)) dγ (r1) . . . dγ (rm). (3.23)

We consider

W ′ =∫

f (r1, . . . , rm) dγ (r1) . . . dγ (rm)U0, (3.24)

where

U0 = EhU(z√q + h1

√1 − q, . . . , z

√q + hm

√1 − q). (3.25)

We claim that

Eh(W −W ′)2 � K(m)

RC ′ 2. (3.26)

To see this, one expends the square and uses the fact that

Eh(U(ξ(r1), . . . , ξ(rm))) = U0

and

Eh(U(ξ(r1), . . . , ξ(rm))U(ξ(rm+1), . . . , ξ(r2m))) = U 20

provided all of r1, . . . , r2m are distinct. We then proceed in a similar manner with S

to see that as R → ∞, µ′ → µ, then ψ(0) approaches the second term of (3.2). ✷

4. Cavity Method (Sort of)

We now have the tools to prove Theorem 1.3. Using symmetry between the sites,we have

E

⟨(σ 1 · σ 2

N− q

)2⟩= E〈f 〉, (4.1)

90 MICHEL TALAGRAND

where

f =(

σ 1 · σ 2

N− q

)(σ 1

Nσ 2N − q). (4.2)

Let us define, for a parameter 0 � t � 1,

Sk,t (σ ) = 1√N

∑i�N−1

σigki +

√t σNgk√N

. (4.3)

There and in the rest of the section we write gk = gkN . Consider a centered

Gaussian r.v. Y , independent of all the other r.v. considered previously and suchthat

EY 2 = q, (4.4)

where q, q are the solutions of (1.13), (1.14). Then, by (1.13), we have

q = Eth2Y. (4.5)

We consider the Hamiltonian HN,M,t given by

−HN,M,t (σ ) =∑k�M

u(Sk,t (σ ))+√1 − t σNY. (4.6)

We denote by 〈·〉t Gibbs’ measure relative to this Hamiltonian, so that 〈·〉1 = 〈·〉.We consider

ϕ(t) = E〈f 〉t , (4.7)

so that

ϕ(1) = E〈f 〉. (4.8)

It is useful to note that

〈f 〉t = 〈f Et 〉〈Et 〉 , (4.9)

where

Et (σ1, σ 2) = exp

∑��2

(u(Sk,t (σ�))− u(Sk(σ

�))+ σ �N

√1 − t Y ).

We have

ϕ(0) = 〈f 〉0 = 1

NE〈(σ 1

Nσ2N − q)2〉0 + 1

NE〈f ′(σ 1

Nσ2N − q)〉0, (4.10)

where

f ′ = 1

N

∑i�N−1

σ 1i σ

2i − q.


Looking at the form of HN,M,0, one sees that

〈f ′(σ 1Nσ

2N − q)〉0 = 〈f ′〉0(th

2Y − q)

and that 〈f ′〉0 is independent of Y . Thus by (4.5) we have

E〈f ′(σ 1Nσ

2N − q)〉0 = 0

and by (4.10),

|ϕ(0)| � 4

N. (4.11)

We write

|ϕ(1)− ϕ(0)| � sup0<t<1

|ϕ′(t)|. (4.12)

Looking at (4.8), (4.11), (4.12), we see that if we prove that

∀t, 0 < t < 1, |ϕ′(t)| � 12E

⟨(σ 1 · σ 2

N− q

)2⟩+ L

N, (4.13)

then we obtain

E

⟨(σ 1 · σ 2

N− q

)2⟩� L

N,

which proves Theorem 1.2. So, we turn to the proof of (4.13). Using (4.9), we seethat

ϕ′(t) = I + II + III,

where, writing S�k,t for Sk,t (σ

�),

I = 1

2√tN

∑��2

∑k�M

Egk〈f σ �Nu′(S�

k,t )〉t ,

II = − 1√tN

∑k�M

Egk〈f σ 3Nu′(S3

k,t )〉t ,

III = 1

2√

1 − t

(2E(Y 〈f σ 3

N〉t )−∑��2

E(Y 〈fσ �N 〉t )

).

To make sense of these expressions, one has to integrate by parts, using (2.1). Thisis another cumbersome but straightforward computation. We get

I = 1

2N

∑��2

∑k�M

E〈f (u′ 2 + u′′)(S�k,t )〉t +

+ 1

N

∑k�M

E〈f σ 1Nσ

2Nu

′(S1k,t )u

′(S2k,t )〉t −

− 1

N

∑��2

∑k�M

E〈f σ �Nσ

3Nu

′(S�k,t )u

′(S3k,t )〉t .

92 MICHEL TALAGRAND

By symmetry between replicas,

I = α

2

∑��2

E〈f (u′ 2 + u′′)(S�M,t )〉t +

+ αE〈f σ 1Nσ 2

Nu′(S1

M,t )u′(S2

M,t )〉t −− α

∑��2

E〈f σ �Nσ 3

Nu′(S1

M,t )u′(S3

M,t )〉t . (4.14)

Similarly, we have

II = −αE〈f (u′ 2 + u′′)(S3M,t )〉t −

− α∑��2

E〈f σ �Nσ

3Nu

′(S�M,t )u

′(S3M,t )〉t +

+ 3αE〈f σ 3Nσ 4

Nu′(S3

M,t )u′(S4

M,t )〉t . (4.15)

Also,

III = −q

(E〈f σ 1

Nσ2N 〉t − 2

∑��2

E〈f σ �Nσ

3N 〉t + 3E〈f σ 3

Nσ4N〉t). (4.16)

In (4.16), the occurrence of q is of course from (4.4).To bring out the dependence of 〈·〉t in SM,t , we introduce the Hamiltonian.

−HN,M−1,t =∑

k�M−1

u(Sk,t (σ ))+√1 − t σNY (4.17)

and we denote by 〈·〉t,1 integration with respect to the corresponding Gibbs mea-sure. We have identities such as

〈f (u′ 2 + u′′)(S�M,t )〉t

= 〈f (u′ 2 + u′′)(S�M,t ) exp

∑��2 u(S

�M,t )〉t,1

〈exp∑

��2 u(S�M,t )〉t,1

. (4.18)

We have developed in Section 3 the tools to evaluate the expectation of such aquantity given 〈·〉t,1. Indeed, the right-hand side of (4.18) is of the type (2.10) form = 2, J = �N , µ the Gibbs measure relative to Hamiltonian (4.17), g(σ ) =SM,t (σ ), U(x1, x2) = (∂2/∂x2

� )F (x1, x2),

F(x1, x2) = V (x1, x2) = exp(u(x1)+ u(x2)).

Assuming (1.11) and

|u(�)| � D′ ∀� � 4, (4.19)

we see that in Theorem 3.1 we can take

B = eLD, C = eLD, C ′ = D′LeLD.


We note that

Eg(σ 1)g(σ 2) = 1

N

∑i�N−1

σ 1i σ

2i + t

Nσ 1i σ

2i ,

so that∣∣∣∣Eg(σ 1)g(σ 2)− σ 1 · σ 2

N

∣∣∣∣ � 1

N

and

|Eg(σ 1)g(σ 2)− q| �∣∣∣∣σ 1 · σ 2

N− q

∣∣∣∣ + 1

N,

|Eg(σ )2 − 1| � 1

N.

Thus

〈(Eg(σ 1)g(σ 2)− q)2〉t,1� 2

N2+ 2

⟨(σ 1 · σ 2

N− q

)2⟩t,1

� 2

N2+ eLD

⟨(σ 1 · σ 2

N− q

)2⟩t

, (4.20)

〈(Eg(σ )2 − 1)2〉t,1 � 1

N2. (4.21)

To control the term A of (3.3), we use the following

LEMMA 4.1. If αD � L, then if N � N0 we have

G⊗2t

({σ , σ ′; |Eg(σ )g(σ ′)| � 1

94

})� exp

(− N

L

), (4.22)

where Gt is Gibbs’ measure relative to the Hamiltonian (4.17).

There is no E in (4.22). This bound is true for all realizations of the disorder.

Proof. If N � N0, we have

|Eg(σ )g(σ ′)| � 1

96⇒∣∣∣∣ 1

N

∑i�N−1

σiσ′i

∣∣∣∣ � 10−2. (4.23)

Let

W ={(σ , σ ′) ∈ �2

N ;∣∣∣∣ 1

N

∑i�N−1

σiσ′i

∣∣∣∣ � 10−2

}.

94 MICHEL TALAGRAND

We have∑(σ ,σ ′)∈W

exp(−HN,M,t (σ )−HN,M,t (σ′)) � ch2Y exp 2MD cardW,

since |∑k�M u(Sk,t (σ ))| � MD. Also,∑σ

exp(−HN,M,t (σ )) � 2Nch Y exp(−MD)

and thus

G⊗2t (W) � exp(4MD)2−2Ncard W.

Now, it is well known that

card

{σ ∈ �N ;

∣∣∣∣ 1

N

∑i�N

σi

∣∣∣∣ � u

}� 2N+1 exp

(−Nu2

2

)

and this implies that

card W � 22N+1 exp

(−(N − 1)

10−4

2

), (4.25)

which, together with (4.24), implies the result. ✷We are now ready to apply (3.2) to the quantity (3.17). We notice that

Lα expLD � 1 implies that q � 1/96. We then find, taking expectations in (3.2)that this quantity is

Ez

Eh(u′ 2 + u′′)(z√q + h

√1 − q) exp u(z

√q + h

√1 − q)

Eh exp u(z√q + h

√1 − q)

××E〈f 〉t,1 + R, (4.26)

where

|R| � LeLD

((E〈f 2〉tE

⟨(σ 1 · σ 2

N− q

)2⟩t

)1/2

+ L

N

)+

+ LeLDD′ 3 exp

(−N

L

)

� LeLD

(E

⟨(σ 1 · σ 2

N− q

)2⟩t

+ L

N

)+ LeLDD′ 3 exp

(−N

L

). (4.27)

We proceed in a similar fashion for all the terms of I and II. We see that byintegration by parts, (1.14) means that

q = αEz

Ehu′(z√q + h1

√1 − q)u′(z√q + h2

√1 − q)E

EhE,


where

E = exp∑��2

u(z√q + h�

√1 − q).

We then see that

I + II = q(E〈f σ 1Nσ 2

N〉t,1 − 2∑��2

E〈f σ �Nσ

3N〉t,1 +

+ 3E〈f σ 3Nσ 4

N〉t,1)+ αR, (4.28)

where R is as in (4.23). Since, for a function ξ on �mN , we have

〈ξ 〉t =〈ξ exp

∑��m u(SM,t (σ

�))〉t,1〈exp

∑� � mu(SM,t (σ �))〉t,1 ,

we can use again (3.2) to see that in fact

I + II = q

(E〈f σ 1

Nσ2N〉t − 2

∑��2

E〈f σ �Nσ 3

N〉t +

+ 3E〈f σ 3Nσ 4

N〉t)+ αR

= −III + αR. (4.29)

Thus, we have proved that

|ϕ′(t)| � LαeLD

(E

⟨(σ 1 · σ 2

N− q

)2⟩t

+ 1

N

)+

+ LαeLDD′ 3 exp

(−N

L

). (4.30)

This looks very much like (4.13), except that on the right-hand side we have 〈·〉trather than 〈·〉. Quite naturally, we set

ξ(t) = E

⟨(σ 1 · σ 2

N− q

)2⟩t

(4.31)

and we try to compare ξ(t) and ξ(1). We compute ξ ′(t), which is given by the sameexpression as ϕ′(t), but where now

f (σ 1, σ 2) =(

σ 1 · σ 2

N− q

)2

.

To each term of I + II + III we now apply (2.16) rather than (3.2) and we get

ξ ′(t) � LαeLD

[ξ(t)+ L

N

]+ LαeLDD′ 3 exp

(− N

L

).

96 MICHEL TALAGRAND

Thus, if LαeLD � 1, D′ � exp(N/L), this becomes

ξ ′(t) � ξ(t)+ L

N

and thus

ξ(t) � Lξ(1)+ L

N.

Going back to (4.30), this gives

|ϕ′(t)| � LαeLD

(E

⟨(σ 1 · σ 2

N− q

)2⟩+ 1

N

)+

+ LαeLDD′ 3 exp

(−N

L

)

and if D′ � exp(N/L), LαeLD � 1, this becomes (4.13). We have proved Theo-rem 1.2.

5. Proof of Theorem 1.1

This proof relies upon an approximation procedure. We will approximate u by afunction v such that v satisfies (1.17), and such that the Gibbs’ measures corre-sponding to u and v respectively are very close in some sense.

We consider a parameter a to be chosen later. It is elementary that there existsa function ϕ supported by [−a, a], ϕ � 0, such that∫

ϕ(x) dx = 1, (5.1)

∀� � 10, |ϕ(�)| � L

a�+1. (5.2)

We define v by

ev = ϕ ∗ eu. (5.3)

Thus, (1.11) implies that

|v| � D. (5.4)

Also,

∀� � 10, |v(�)| � LeLD

a�. (5.5)


To prove (5.3), we note that

v′ = ϕ′ ∗ eu

ϕ ∗ eu,

so that (e.g.)

|v′| � ‖ϕ′‖1e2D � Le2D/a.

LEMMA 5.1. If b ∈ R and g is centered Gaussian we have

|E(eu(g+b) − ev(g+b))| � eDa√Eg2

. (5.6)

Proof. Consider V such that V ′(x) = eu(x). Then (ϕ ∗ V )′ = ϕ ∗ V ′ = ev .Moreover, since |V ′| � eD , and since the support of ϕ is contained in [−a, a], wehave

|V − ϕ ∗ V | � aeD. (5.7)

Now, since (V − ϕ ∗ V )′ = eu − ev , integration by parts show that

Eg2E(eu(g+b) − ev(g+b)) = E(g(V (g + b)− ϕ ∗ V (g + b)))

so that, using (5.7), we see that

Eg2|E(eu(g+b) − ev(g+b))| � aeDE|g| � aeD(Eg2)1/2

and this implies (5.6). ✷We recall the notation Sk(σ ) = N−1/2

∑i�N gk

i σi . The key point is as follows.

LEMMA 5.2. Consider a subset A of �N . Then

E

(∑σ∈A

(e∑

k�M u(Sk(σ )) − e∑

k�M v(Sk(σ ))))2

� Le2MD(cardA+ a2NM2(card A)2). (5.8)

Proof. The left-hand side of (5.8) is∑σ ,σ ′∈A

E(B(σ )B(σ ′)), (5.9)

where

B(σ ) = e∑

k�M u(Sk(σ )) − e∑

k�M v(Sk(σ )).

If σ = σ ′ or σ = −σ ′ we use the trivial bound

|B(σ )| � 2eMD

98 MICHEL TALAGRAND

to obtain

|E(B(σ )B(σ ′))| � 4e2MD.

Next, we fix σ , σ ′ with σ �= σ ′, σ �= −σ ′. Without loss of generality, we assumeσ1 = σ ′

1, σ2 = −σ ′2. To study E(B(σ )B(σ ′)) we first study E0(B(σ )B(σ ′)), where

E0 denotes conditional expectation given gki , k � M, 3 � i � N . In other words,

we integrate only in gki , k � M, i = 1, 2.

We note the fact that the 2M variables

gk = σ1gk1 + σ2g

k2 and g′k = σ ′

1gk1 + σ ′

2gk2

are all independent, so that

E0(B(σ )B(σ ′)) = E0(B(σ ))E0(B(σ ′)). (5.10)

Now

E0(B(σ )) =∏k

Xk −∏k

Yk, (5.11)

where

Xk = E0eu(Sk(σ )); Yk = E0ev(Sk(σ )).

Using Lemma 5.1 with g = gk/√N , b = N−1/2

∑i�3 σig

ki , we see that

|Xk − Yk| � aN1/2eD.

Since |Xk| � eD, we see from (5.10) that

E0(B(σ )) � aN1/2MeMD.

Combining with (5.10) this finishes the proof. ✷We write

−HN,M,u(σ ) =∑k�M

u(Sk(σ )),

ZN,M,u =∑

σ

exp(−HN,M,u(σ ))

and similar obvious notation for v.

COROLLARY 5.3. For each ε > 0,

P

(∣∣∣∣∑σ∈A

(e−HN,M,u(σ ) − e−HN,M,v (σ ))

∣∣∣∣ � ε card A

)

� Le2MD

ε2

(1

card A+ a2NM2

). (5.12)


Proof. Use (5.8) and Markov’s inequality. ✷We now choose a = 2−2rN , where r > 0 is small enough that 21r < 1/L,

where L is the constant of (1.17). Then (5.5) shows that for N � N(D), v satisfies(1.17) if LαD � 1. Next, in (5.12) we use ε = exp(−MD − rN). The right-handside of (5.12) is then bounded by

Le4MD−rN

(e3rN

card A+NM2e−rN

). (5.13)

Next we observe that ZN,M,u � 2Ne−MD. If η1, η2 = ±1, using (5.12) forA = �N and for A = {σ ; σ1 = η1, σ2 = η2} elementary manipulations then showthat

E|GN,M,u({σ1 = η1, σ2 = η2})−GN,M,v({σ1 = η1, σ2 = η2})|� LNM2e4MD−rN � Le−rN/2 (5.14)

provided LαD � 1.Writing σ · σ ′ =∑

i�N σiσ′i and using symmetry between sites, we see that

E

⟨(σ · σ ′

N− q

)2⟩= 1

N+ N − 1

NE〈σ1σ2〉2 + q2 − 2qE〈σ1〉2

and combining with (5.14) we get∣∣∣∣E⟨(

σ · σ ′

N− qu

)2⟩u

− E

⟨(σ · σ ′

N− qv

)2⟩v

∣∣∣∣� L|qu − qv| + Le−rN/2. (5.15)

We will leave to the reader to check (using (5.6)) that |qu − qv| � Le−rN/2.Inequality (5.15) makes it obvious that Theorem 1.1 follows from Theorem 1.2.

References

[GD] Gardner, E. and Derrida, B.: Optimal storage properties of neural network models, J. Phys.A 21(1) (1988), 271–284.

[HKP] Hertz, J., Krogh, A. and Palmers, R. C.: Introduction to the Theory of Neural Computation,Addison-Wesley, 1991.

[MPV] Mézard, M., Parisi, G. and Virasiro, M.: Spin Glass Theory and Beyond, World Scientific,Singapore, 1987.

[T1] Talagrand, M.: Huge random structures and mean field models for glasses, In: Proc. BerlinInternat. Congr. Document. Math, Extra Volume I (1988), 507–536.

[T2] Talagrand, M.: Intersecting random half spaces: Towards the Derrida–Gardner formula,Ann. Probab. 28 (2001), 725–758.

[T3] Talagrand, M.: Mean Field Models for Spin Glasses: A First Course, Saint Flour SummerSchool in Probability 2000, to appear in Springer Lecture Notes in Math.


101

Unitary Correlations and the Fejér Kernel

Dedicated to Harold Widom on his 70th birthday

DANIEL BUMP, PERSI DIACONIS and JOSEPH B. KELLERDepartment of Mathematics, Stanford University, Stanford, CA 94305, U.S.A.

(Received: 28 March 2002)

Abstract. Let M be a unitary matrix with eigenvalues tj , and let f be a function on the unit circle.Define Xf (M) = ∑

f (tj ). We derive exact and asymptotic formulae for the covariance of Xf

and Xg with respect to the measures |χ(M)|2 dM where dM is Haar measure and χ an irreduciblecharacter. The asymptotic results include an analysis of the Fejér kernel which may be of independentinterest.

Mathematics Subject Classifications (2000): 15A52, 34E05.

Key words: random matrix theory, Fejér kernel, unitary group.

1. Introduction

Random matrix theory uses the eigenvalues of typical large matrices as modelsof a wide variety of natural phenomena from nuclear spectra to the zeros of theRiemann zeta function. Surveys of this area may be found in [17, 24] or [9].

We study the eigenvalues of typical unitary matrices M. These are n points tjon T, the unit circle. Sometimes we will write tj = eiθj with 0 � θj < 2π .The distribution of the traces of powers of M is studied in [4], the number ofeigenvalues in an interval is studied in [26], and the log characteristic polynomialis studied in [13, 14] and [11]. These are all examples of additive functions of M,that is, functions of the form

Xf (M) =n∑

j=1

f (tj ),

where f : T → C is a function. The limiting behavior of such functionals is studiedin the papers cited above as well as in [3, 20, 21], where a variety of central limittheorems are proved.

In this paper we study the variance and covariance of Xf and Xg . We are inter-ested not merely in the limiting behavior, but in the behavior for n of moderate size.

We will normalize Haar measures∫

Tdt and

∫U(n)

dM so that the compactgroups T and U(n) have volume 1. The convolution of two functions on T is definedby

(f ∗ g)(x) =∫

T

f (t)g(xt−1) dt.

102 DANIEL BUMP ET AL.

The Fejér kernel

Kn(t) =n−1∑

k=−(n−1)

(1 − |k|

n

)tk = sin(nθ2 )

2

n sin( θ2 )2, t = eiθ . (1)

Fejér introduced this kernel to prove that Fourier coefficients determine the func-tion at a point under suitable conditions. See [15] for background and [27],pp. 88–90, for the classical theory.

If � and � are functions on U(n), define the covariance

Cov(�,�) =∫

U(n)

�(M)�(M) dM − ��,

where the mean value

� =∫

U(n)

�(M) dM.

Let g(t) = g(t−1).

THEOREM 1. For f, g ∈ L2(T),

Cov(Xf ,Xg) = n(f ∗ g)(1) − n(f ∗ g ∗ Kn)(1). (2)

It is striking that this covariance only depends on f ∗ g.We will give two proofs of Theorem 1, in Sections 2 and 3. We will see in

Section 2 that Theorem 1 is equivalent to an icon of random matrix theory, a for-mula of Dyson for the pair correlation function. We will generalize Theorem 1 inTheorem 4 below to obtain covariances with respect to the measure |χ(M)|2 dM,where χ is an irreducible character of U(n) by a very similar formula.

The Fejér kernel is a Dirac sequence. This means that Kn � 0,∫

TKn(t) dt = 1

and Kn → 0 uniformly on any compact subset of T excluding the point 1. Con-sequently the sequence of trigonometric polynomials φ ∗ Kn → φ uniformly forcontinuous functions φ on T. Taking φ = f ∗ g we see that the covariance (2) maybe interpreted as the error in the approximation of φ by φ ∗ Kn.

Suppose that ck and dk are the Fourier coefficients of f and g, so that f (t) =∑ckt

k and g(t) = ∑dkt

k. Recalling that convolution of functions corresponds tomultiplication of the Fourier coefficients, we may write (2) in the equivalent form:

Cov(Xf ,Xg) = n

∞∑k=−∞

ckd−k −n−1∑

k=−(n−1)

(n − |k|)ckd−k. (3)

As an example, we consider the case where f and g are the characteristicfunctions of two intervals. Thus Xf and Xg count the number of eigenvalues ineach interval. Wieand [26] showed that in the limit as n → ∞, these functions areuncorrelated unless the intervals share an endpoint. She found that if they share

UNITARY CORRELATIONS AND THE FEJER KERNEL 103

a left or right endpoint, then there is a positive limiting correlation, but if the leftendpoint of one interval is a right endpoint of the other, then there is a negativelimiting correlation.

Meanwhile Rains [19] also considered the number of eigenvalues in an interval;he found the complete asymptotic expansion for the variance. It is possible to gofrom Rains’ asymptotic results on the variance of the number of eigenvalues inan interval I to an asymptotic result on the covariance of the number of eigen-values in two different intervals I and J . The work involved in this process isnot entirely trivial, since the results of [19] would need to be applied to severalintervals: if K is a connected component of the symmetric difference I�J of Iand J , then one needs to know the variances in the number of eigenvalues in thesix intervals I , J , K, I�K, J�K and I�(J�K) = (I�J )�K. By an inclusionexclusion process, one may infer the covariance of the number of eigenvalues inthe intervals I and J .

We can consider this matter from our point of view using Theorem 1, recoveringand extending these results of Wieand and Rains. As an example, let 0 � ω � 2π ,and let f and g be the characteristic functions of the images under θ → eiθ of theintervals [0, ω] and [θ, ω + θ], respectively. We have

ck = eikθ dk ={

12πik (1 − e−ikω), if k �= 0,ω

2π , if k = 0.

Let φ = f ∗ f . Thus

φ(eiθ ) = max(ω − θ, 0) + max(ω + θ − 2π, 0)

2π. (4)

Because

(f ∗ g)(1) = φ(eiθ ) and (f ∗ g ∗ Kn)(1) = (φ ∗ Kn)(eiθ ),

Theorem 1 shows that Cov(Xf ,Xg) = n(φ − φ ∗ Kn)(eiθ ), and it is this that westudy. The graph of φ(eiθ ) in the case ω = π/2 is shown in Figure 1.

Figure 1. φ(eiθ ) as a function of θ for ω = π/2.


Figure 2. The function n(φ − φ ∗ Kn) when n = 42.

The series (3) simplifies to

Cov(Xf ,Xg)

= n(φ − φ ∗ Kn)(eiθ )

= nφ(eiθ ) − n

(ω

2π

)2

−n−1∑k=1

(n − k)

π2k2(1 − cos(kω))cos(kθ). (5)

This formula is suitable for numerical computation. For n = 42 and ω = π/2,Figure 2 shows the graph of this quantity as a function of θ .

Figure 2 displays the covariance between the number of eigenvalues of a ran-dom unitary matrix in two given intervals of length π/2 as one of the intervalsslides around the circle. We see that the places where (5) has the largest magnitudeare the locations of the discontinuities in the derivative φ′. We will justify thisqualitative observation in Section 4 by an asymptotic analysis of φ − Kn ∗ φ fora function φ which (like this one) is continuous, but whose derivative has jumpdiscontinuities.

Let Ci and Si be the cosine and sine integrals,

Ci(x) = γ + log(x) +∫ x

0

cos(t) − 1

tdt

= −∫ ∞

x

cos(t)

tdt, |arg(x)| < π, (6)

where γ = 0.57721 . . . is Euler’s constant, and

Si(x) =∫ x

0

sin(t)

tdt = π

2−

∫ ∞

x

sin(t)

tdt, |arg(x)| < π. (7)

The asymptotic expansion of Ci as x → ∞, obtained from the last expressionin (6) by integration by parts, is

Ci(x) ∼ sin(x)

x

(1 − 2!

x2+ 4!

x4− · · ·

)− cos(x)

x2

(1 − 3!

x2+ 5!

x4− · · ·

). (8)

Similarly

Si(x) ∼ π

2− sin(x)

x2

(1 − 3!

x2+ 5!

x4− · · ·

)−

− cos(x)

x

(1 − 2!

x2+ 4!

x4− · · ·

). (9)


Figure 3. The function �5.

If |θ | � π , let

�n(θ) =

2(1 + γ + log(2n)), if θ = 0,

2 Ci(n|θ |) − 2 log sin( |θ |

2

) + 2 cos(nθ) − nπ |θ | + 2nθ Si(n|θ |),otherwise.

(10)

By (8) and (9) this function is continuous at θ = 0. We make �n into a 2π peri-odic function. The function �n is ‘spiky’, increasingly so as n increases. Indeed,(8) and (9) show that

limn→∞�n(θ) = −2 log sin(|θ |/2), (11)

n → ∞. The graph of �5 is shown in Figure 3.We can now describe the results of our asymptotic analysis of φ − Kn ∗ φ,

which are given in Theorems 9 and 10 below. The most important and interestingcontributions to these come from the jump discontinuities in the derivative of φ.Indeed, we will show that if θ → φ(eiθ ) has a jump discontinuity at θ = θ0,then the asymptotic form of n(φ − Kn ∗ φ)(eiθ ) contains a constant multiple of�n(θ − θ0).

When φ is the function (4), the derivative jumps at θ = 0, ω and 2π − ω, andTheorem 10 shows that

Cov(Xf ,Xg) = n(φ − Kn ∗ φ)(θ)

= 1

2π2�n(θ) − 1

4π2�n(θ − ω) −

− 1

4π2�n(θ − 2π + ω) + O(n−1)

uniformly in θ . In practice this approximation is quite good. For n = 42, the graphof the approximation is indistinguishable from Figure 2, for its values agree withthose of the original function to about five decimal places. The largest error is lessthan 1.4 × 10−5.


Figure 4. The limiting covariance as n → ∞ for Figure 1.

Theorem 1, together with (11) and Theorem 9 show that as n → ∞ the co-variance Cov(Xf ,Xg) tends to a limiting distribution which is singular at thediscontinuities of the derivative of φ. In the example of Figure 1, (11) shows thatthe limiting covariance equals

1

2π2log

∣∣∣∣∣sin(θ−π/2

2

)sin

(θ−3π/2

2

)sin2(θ/2)

∣∣∣∣∣.The graph of this function is shown in Figure 4.

We will generalize Theorem 1 by obtaining covariances with respect to theprobability measure |χλ(M)|2 dM, where λ is a partition of length � n, and χλ isthe character of the irreducible representation indexed by λ, defined in (16) below.Denoting this covariance as Covλ, we will show in Theorem 4 that if λ is fixed,there is a Dirac sequence Kn,λ such that

Covλ(Xf ,Xg) = n(f ∗ g − f ∗ g ∗ Kn,λ)(1).

For example, if λ = (1), that is, the partition (1, 0, 0, . . .), then χλ(M) = tr(M)

is just the character of the standard representation of U(n). When n = 42 and φ isthe function (4) with ω = π/2, the graph of this covariance is shown in Figure 5.

We will show in Theorem 11 that the functions �n can be generalized to func-tions �n,λ, giving in a similar way the asymptotic forms of Covλ(Xf ,Xg), at leastwhen f and g are piecewise linear and continuous. We do not work out the exactform of �n,λ, though we give the interested reader enough information to do so inthe proofs of Theorems 3 and 11. We find that �n,λ(θ) − �n(θ) is a trigonometricpolynomial independent of n. For example, in the case where λ = (1), we find that

�n,λ(θ) − �n(θ) = 2 cos(θ).

This result shows that if λ is fixed and n → ∞ the covariance Covλ(Xf ,Xg)

tends to a limiting distribution, which is finite except at the discontinuities of thederivative of φ.


Figure 5. The function n(φ − φ ∗ Kn,λ) when n = 42, λ = (1).

There has been some related work since the circulation of a preliminary ver-sion of this paper. Hughes [10] derives related approximations in comparing thevariance of the number of eigenvalues and zeta zeros in matching intervals. In thematter of Fejér asymptotics, Pinsky [18] recognized the correction term in The-orem 7 as the Hilbert transform of f ′. He derives L2 convergence results fora variety of summability kernels and extends his results from the circle to the line.Taylor [23] gives a version of Pinsky’s results for a function on an n-dimensionalRiemannian manifold expanded in eigenfunctions of the Laplacian. He furtherextends our results to uniform asymptotics for piecewise smooth f with a sim-ple jump across a smooth hypersurface. We thank these authors for keeping usinformed.

2. Pair Correlation

The formula (2) is equivalent to a basic formula of unitary statistics, Dyson’s for-mula for the correlation function for unitary eigenvalues. The correlation functionswere found by Dyson in part III of [5]; Dyson gave a proof and a generalization toother ensembles in [6]; see also [17] and [24] for different proofs and extensions.

The m-level correlation Rm(t1, . . . , tm) measures the density that t1, . . . , tm arethe eigenvalues of a Haar random unitary matrix. Concretely, if f (t1, . . . , tm) isa test function on T

m, then∫Tm

Rm(u1, . . . , um)f (u1, . . . , um) du1 . . . dum

=∫

U(n)

∑∗f (ti1, . . . , tim) dM, (12)

where the sum on the right is over the n!/(n − m)! different m-tuples (i1, . . . , im)

where the ij are pairwise distinct integers between 1 and n and where t1, . . . , tn arethe eigenvalues of M.

Dyson found that

Rm(t1, . . . , tm) = det(sn(θj − θk))j,k, tj = eiθj , (13)


where

sn(θ) ={

sin(nθ/2)sin(θ/2) , θ �= 0;n, θ = 0.

In the case m = 2 (the only case we need), this amounts to

R2(t1, t2) = n2 − nKn(t1t−12 ). (14)

Proof of Theorem 1. If ti are the eigenvalues of M, denote

&(M) =n∑

j=1

f (tj )g(tj ).

Evidently

Xf (M)Xg(M) − &(M) =∑j �=k

f (tj )g(tk).

Using (12) we have∫U(n)

(Xf (M)Xg(M) − &(M)) dM

=∫

T2R(u1, u2)f (u1)g(u2) du1 du2

=∫

T2(n2 − nKn(u1u

−12 ))f (u1)g(u2) du1 du2.

The left-hand side here equals∫U(n)

Xf (M)Xg(M) dM − n(f ∗ g)(1),

and the right side equals

Xf Xg − n(f ∗ g ∗ Kn)(1).

Comparing these gives (2). ✷Let λ be a partition, that is, a decreasing sequence λ1 � λ2 � λ3 � · · · of

integers such that λj = 0 for j sufficiently large. The largest l such that λl �= 0 iscalled the length l(λ) of λ. If the length of λ is � n, let

sλ(t1, . . . , tn) =

∣∣∣∣∣∣∣∣∣

tλ1+n−11 t

λ1+n−12 · · · tλ1+n−1

n

tλ2+n−21 t

λ2+n−22 · · · tλ2+n−2

n...

...

tλn1 t

λn2 · · · tλnn

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

tn−11 tn−1

2 · · · tn−1n

tn−21 tn−2

2 · · · tn−2n

......

1 1 · · · 1

∣∣∣∣∣∣∣∣∣

. (15)


This is the Schur function as defined in [16], (3.1) on p. 30. See [16], volume 2of [22] and [1] for background on the Schur functions and related representationtheory.

If t1, . . . , tn are the eigenvalues of M ∈ U(n), then

χλ(M) = sλ(t1, . . . , tn) (16)

is the character of an irreducible representation of U(n), denoted sλ(M) in [1].(This is essentially the Weyl character formula.) With dM Haar measure,|χλ(M)|2 dM defines a probability measure on U(n). We now investigate the m-le-vel correlation function Rm,λ(t1, . . . , tm) with respect to the measure |χλ(M)|2 dM.We define this by analogy with (12) by asking that∫

Tm

Rm,λ(t1, . . . , tm)f (t1, . . . , tm) dt1 . . . dtm

=∫

U(n)

∑∗f (ti1, . . . , tim)|χλ(M)|2 dM. (17)

We will prove a formula generalizing Dyson’s (13) for Rm,λ. To motivate this,rewrite (13) this way:

Rm(t1, . . . , tm) = det(AA∗),

where A is the (nonsquare) matrix

1 t1 t21 · · · tn−1

1

......

...

1 tm t2m · · · tn−1

m

and A∗ is its conjugate transpose. The Hermitian matrix AA∗ is not equal to thesquare matrix on the right side of (13), but is conjugate to it by a diagonal matrix.

THEOREM 2. We have

Rm,λ(t1, . . . , tn) = det(AλA∗λ), (18)

with

Aλ =

t−λ11 t

1−λ21 · · · t

−λn+n−11

......

...

t−λ1m t1−λ2

m · · · t−λn+n−1m

.

Proof. We may prove this formula as follows. First suppose that m = n. Thenthe matrix Aλ is square, and

det(AλA∗λ) = |det(A∗

λ)|2.


We must show that the function on the right side of (18) satisfies (17). By the Weylintegration formula, if we substitute |det(A∗

λ)|2 for Rm,λ, on the left side of (17) weobtain

∫Tn

∣∣∣∣∣∣∣∣∣∣∣

tλ11 t

λ12 · · · tλ1

n

tλ2−11 t

λ2−12 · · · tλ2−1

n

......

tλn−n+11 t

λn−n+12 · · · tλn−n+1

n

∣∣∣∣∣∣∣∣∣∣∣

2

f (t1, . . . , tn) dt1 . . . dtn.

Let

F(t1, . . . , tn) =∑σ∈Sn

f (tσ(1), . . . , tσ (n))

be the symmetrization of f . The Vandermonde identity and the symmetry of theSchur function show that the last expression equals

1

n!∫

Tn

∣∣∣∣∣∣∣∣∣

tλ1+n−11 t

λ1+n−12 · · · tλ1+n−1

n

tλ2+n−21 t

λ2+n−22 · · · tλ2+n−2

n...

...

tλn1 t

λn2 · · · tλnn

∣∣∣∣∣∣∣∣∣

2

∣∣∣∣∣∣∣∣∣

tn−11 tn−1

2 · · · tn−1n

tn−21 tn−2

2 · · · tn−2n

......

1 1 · · · 1

∣∣∣∣∣∣∣∣∣

2 F(t1, . . . , tn)∏i<j

|ti − tj |2 dt1 . . . dtn.

Now using (15) and the Weyl integration formula (Goodman and Wallach [8],p. 343), we obtain the right side of (17).

Therefore (18) is true when m = n. The case m < n follows by the same argu-ment as in [16], pp. 195–196. The downward induction is based on Theorem 5.2.1on p. 89 of [16], where in the case at hand the function f (x, y) on T is

f (x, y) =n∑

j=1

(xy−1)j−1−λj .

The other ingredient of the proof is the analog of Equation (5.1.2) of [16], whichfor us is the formula

Rm,λ(x1, . . . , xm) = 1

(n − m)!∫

Tn−m

Rn,λ(x1, . . . , xn) dxm+1 . . . dxn.

It is straightforward to deduce this from (17) by taking a test function f (t1, . . . , tn)

which only depends on t1, . . . , tm. ✷


If n � l(λ), define

Kn,λ(t) = 1

n

∣∣∣∣∣n∑

j=1

tλj−j

∣∣∣∣∣2

.

THEOREM 3. (i) The sequence of functions Kn,λ is a Dirac sequence.(ii) There exists polynomials fλ(t) and gλ(t) in t and t−1 such that fλ(t) =

fλ(t−1) and

n(Kn,λ(t) − Kn(t)) = fλ(t) + tngλ(t) + t−ngλ(t−1). (19)

Proof. It is evident that Kn,λ is positive on T.Writing

Kn,λ(t) = 1

n

∑j,k

tλj−λk−j+k,

its mean value on T is 1/n times the number of pairs j, k with λj − j = λk − k.Since λj − j is a decreasing sequence, these pairs occur precisely when j = k, so∫

TKn,λ(t) dt = 1.Next we show that if t �= 1, then Kn,λ(t) → 0; the convergence is uniform on

compact subsets of T − {1}. If l = l(λ) is the length of λ, then

Kn,λ(t) = 1

n

∣∣∣∣∣l∑

j=1

(tλj − 1)t−j + 1 − t−(n+1)

1 − t−1

∣∣∣∣∣2

. (20)

Now∑l

j=1(tλj −1)t−j is independent of n, and (1− t−(n+1))/(1− t−1) is bounded

independently of n if t is bounded away from 1. Hence (20) is O(n−1).We have established that Kn,λ is a Fejér sequence. It follows from (20) that

n(Kn,λ(t) − Kn(t))

=∣∣∣∣∣

l∑j=1

tλj−j − t−j

∣∣∣∣∣2

+l∑

j=1

tλj−j − t−j

1 − t(1 − tn+1) +

+l∑

j=1

t−λj+j − tj

1 − t−1(1 − t−n−1).

Noting that (tλj−j − t−j )/(1 − t) is a polynomial in t and t−1, simplifying thisgives (19). ✷

If � and � are functions on U(n) we will define Covλ(�,�) to be the covari-ance with respect to the measure |χλ(M)|2 dM.


THEOREM 4. We have

Covλ(Xf ,Xg) = n(f ∗ g)(1) − n(f ∗ g ∗ Kn,λ)(1). (21)

Proof. If m = 2, we may rewrite Theorem 2 in the form

R2,λ(t1, t2) = n2 − nKn,λ(t1t−12 ).

The proof of Theorem 4 is now the same as that of Theorem 1. ✷

3. Principal Toeplitz Minors

In this section we reprove Theorems 1 and 4 by an entirely different method.A classical identity of Heine and Szegö expresses Toeplitz determinants as integralsover the unitary group. See Bump and Diaconis [1] for generalizations and appli-cations of this identity. Here we will derive another generalization of this identitywhich contains information equivalent to the m-level correlation function of unitarystatistics.

Let 1 � m � n. If M is any matrix, let Em(M) denote the sum of the(n

m

)principal m × m minors of M. Since this is just the trace of M in the mth exteriorpower representation of U(n), it is invariant under conjugation, so to computeit we may assume M is diagonal. Thus Em(M) = em(t1, . . . , tn), where tj arethe eigenvalues of M, and em is the mth elementary symmetric polynomial in n

variables.If f is a continuous function on T, we may associate with f a continuous

function Uf : U(n) → U(n), namely

Uf

h

t1. . .

tn

h−1

= h

f (t1). . .

f (tn)

h−1,

with h ∈ U(n). Note that this is well defined. If dj are the Fourier coefficients of f ,so that f (t) = ∑

dj tj , let Tn−1 be the n × n Toeplitz matrix

Tn−1(f ) =

d0 d1 · · · dn−1

d−1 d0 · · · dn−2

......

d−(n−1) d−(n−2) · · · d0

.

THEOREM 5. With these notations,∫U(n)

Em(Uf (M)) dM = Em(Tn−1(f )). (22)


Proof. If m = n, this identity may be written∫U(n)

det(Uf (M)) dM = det(Tn−1(f )), (23)

and this is the Heine–Szegö identity. See [1] for a proof and generalizations. Thegeneral case of (22) follows by replacing f by 1 + λf in (23), then expanding andcomparing the coefficients of λm. ✷

One may deduce Theorem 1 from Theorem 5 by an argument we will give belowin the second proof of Theorem 4. To obtain Theorem 4, we need to generalizeTheorem 5.

If λ and µ, are partitions of length � n, let T λ,µ

n−1(f ) be the Toeplitz minor whichis the n × n matrix whose (j, k)th entry is dλj−λk−j+k. It is a minor in a largerToeplitz matrix. The asymptotics of large Toeplitz minors were studied by Bumpand Diaconis [1]. If λ = µ this is a principal minor. These are the ones which wewill need.

THEOREM 6. We have∫U(n)

Em(Uf (M))χλ(M)χµ(M) dM = Em(Tλ,µ

n−1(f )). (24)

Proof. We take λ = µ in Theorem 3 of [1]. The quoted theorem asserts that∫U(n)

det(Uf (M))χλ(M)χµ(M) dM = det(T λ,µ

n−1(f )).

Proceeding as in the proof of Theorem 5 we obtain (24). ✷In our applications of Theorem 6 we will take λ = µ. The special case m = 1

of (24) is worth noting:∫U(n)

Xf (M)|χλ(M)|2 dM = n

∫T

f (t) dt. (25)

Second Proof of Theorem 4. Let

f (t) =∑

cktk and g(t) =

∑dkt

k.

Let M ∈ U(n) have eigenvalues tj . Then

Xf (M)Xg(M) =∑j,k

f (tj )g(tk).


We integrate this with respect to the measure |χλ(M)|2 dM, separating the diagonalterms (j = k) from the terms with j �= k. The diagonal contribution equals theintegral of Xfg(M), which by (25) equals

n

∫T

f (t)g(t) dt = n∑k

ckd−k. (26)

We are left with the integral of the off-diagonal terms∑j �=k

f (tj )g(tk) = E2(Uf+g(M)) − E2(Uf (M)) − E2(Ug(M)).

We evaluate this by means of (24). It equals

E2(Tλ,λn−1(f + g)) − E2(T

λ,λn−1(f )) − E2(T

λ,λn−1(g)).

Let 1 � k � n − 1. If 1 � j � k � n, then Tλ,λn−1(f + g) has a principal minor of

the form∣∣∣∣∣ c0 + d0 cλj−λk−j+k + dλj−λk−j+k

c−(λj−λk−j+k) + d−(λj−λk−j+k) c0 + d0

∣∣∣∣∣ .From this, we subtract the two corresponding minors∣∣∣∣∣ c0 cλj−λk−j+k

c−(λj−λk−j+k) c0

∣∣∣∣∣ +∣∣∣∣∣ d0 dλj−λk−j+k

d−(λj−λk−j+k) d0

∣∣∣∣∣of T λ,λ

n−1(f ) and Tλ,λn−1(g) to obtain

2c0d0 − cλj−λk−j+kd−(λj−λk−j+k) − c−(λj−λk−j+k)dλj−λk−j+k.

Summing these terms gives

(n2 − n)c0d0 −∑

1�j<k�n

(cλj−λk−j+kd−(λj−λk−j+k) +

+ c−(λj−λk−j+k)dλj−λk−j+k)

= (n2 − n)c0d0 −∑

1�j �=k�n

cλj−λk−j+kd−(λj−λk−j+k).

Adding back the diagonal terms (26) gives∫U(n)

Xf (M)Xg(M)|χλ(M)|2 dM

= n

∞∑k=−∞

ckd−k + n(n − 1)c0d0 −∑

1�j �=k�n

cλj−λk−j+kd−(λj−λk−j+k).


By (25) we have

Xf =∫

U(n)

Xf (M)|χλ(M)|2 dM = nc0 and Xg = nd0.

Thus

1

nCovλ(Xf ,Xg) =

∞∑k=−∞

ckd−k −∞∑

k=−∞ρlcld−l ,

where ρl = 1 if l = 0; more generally, it is 1/n times the number of pairs (j, k)

with λj − λk − j + k = l. Evidently∑

ρltl = Kn,λ(t), whence (21). ✷

4. Fejér Asymptotics

As we have noted, the convolution of a function f with the Fejér kernel givesa sequence of approximations to f by trigonometric polynomials. We have ex-pressed the covariance of two additive functions on U(n) as the error in suchan approximation. The class of functions on T which occurs in Theorem 1 con-sists of convolutions of pairs of functions, which in the applications might bepiecewise smooth with jump discontinuities. The convolution of a pair of suchfunctions is then continuous and piecewise smooth but its derivative can have jumpdiscontinuities.

For the analysis of the asymptotics of the convolution of a function f with theFejér kernel, it will be useful to parametrize the circle by the interval (−π, π ]. Wetherefore denote

kn(x) = sin2(nx/2)

n sin2(x/2)= 1 − cos(nx)

2n sin2(x/2), Kn(e

ix) = kn(x). (27)

We have

1

2π

∫ π

−π

kn(x) dx = 1. (28)

The convolution of kn with a 2π -periodic function f (θ) is defined by

(f ∗ kn)(θ) = 1

2π

∫ π

−π

kn(x)f (θ + x) dx. (29)

We rewrite (29), making use of (28), in the form

(f ∗ kn)(θ) = f (θ) + 1

2π

∫ π

−π

kn(x)[f (θ + x) − f (θ)] dx. (30)

First we assume that the derivative f ′(θ) exists. Let

R(x, θ;f ) = f (x + θ) − f (θ) − f ′(θ)x. (31)


THEOREM 7. Let f be a 2π -periodic integrable function such that f ′(θ) exists,and such that R(x, θ;f ) = x2φθ (x) where φθ(x) is integrable as a function of x.Then

(f ∗ kn)(θ) = f (θ) + 1

4πn

∫ π

−π

R(x, θ;f )sin2(x/2)

dx + o(n−1). (32)

If φ′θ (x) exists and is integrable then the error term in (32) is O(n2).Proof. Using (30),

(f ∗ kn)(θ) = f (θ) + f ′(θ)2π

∫ π

−π

xkn(x) dx + 1

2π

∫ π

−π

kn(x)R(x, θ;f ) dx.

The first integral on the right vanishes since xkn(x) is odd. By using the secondexpression in (27), we may write

(f ∗ kn)(θ) = f (θ) + 1

4πn

∫ π

−π


dx −

− 1

4πn

∫ π

−π

x2

sin2(x/2)φθ (x)cos(nx) dx.

By the Riemann–Lebesgue Lemma, the last term is o(n−1). If φ′θ is integrable, then

integration by parts shows that this term is O(n−2). The theorem follows. ✷LEMMA 1. We have

1

2π

∫ π

0xkn(x) dx = 1

nπ(1 + γ + log(2n)) + O(n−2). (33)

Proof. Using the second expression in (27) converts the integral in (33) into

1

4πn

∫ π

0x(1 − cos(nx))

[4

x2+

(1

sin2(x/2)− 1

(x/2)2

)]dx

= 1

nπ

∫ π

0

1 − cos(nx)

xdx + 1

4nπ

∫ π

0x

(1

sin2(x/2)− 1

(x/2)2

)dx −

− 1

4nπ

∫ π

0x

(1

sin2(x/2)− 1

(x/2)2

)cos(nx) dx.

The first integral on the right-hand side is

1

nπ(log nπ + γ − Ci(nπ)),

where by (8) we have Ci(nπ) = O(n−2). As for the second integral,

1

4nπ

∫ π

0x

(1

sin2(x/2)− 1

(x/2)2

)dx = 1 − log(π/2)

nπ.


The last integral is O(n−2), as can be proved by integration by parts, and puttingthese together, we obtain the lemma. ✷

Next we assume that f ′ may be discontinuous at θ = θ0, but that its right andleft derivatives f ′+(θ0) and f ′−(θ0) both exist. Let

f (x + θ0) ={f (θ0) + f ′+(θ0)x + R+(x, θ0;f ), x > 0,

f (θ0) + f ′−(θ0)x + R−(x, θ0;f ), x < 0.

We obtain the asymptotics of (f ∗ kn)(θ) first when θ = θ0, and later when θ isnear to but different from θ0. Let

R(x, θ0;f ) ={R+(x, θ0;f ) if x > 0,

R−(x, θ0;f ) if x < 0.(34)

THEOREM 8. Let θ = θ0, where f ′ has a jump discontinuity. Assume thatR(x, θ0;f ) = x2φθ0(x), where φθ0(x) is integrable as a function of x. Then

(f ∗ kn)(θ0) = f (θ0) + f ′+(θ0) − f ′−(θ0)

nπ(1 + γ + log(2n)) +

+ 1

4πn

∫ π

−π

R(x, θ0;f )sin2(x/2)

dx + o(n−1). (35)

Proof. We have

(f ∗ kn)(θ0) = f (θ0) + 1

2π

∫ 0

−π

kn(x)R−(x, θ0;f ) dx +

+ 1

2π

∫ π

0kn(x)R+(x, θ0;f ) dx +

+ f ′−(θ0)

2π

∫ 0

−π

xkn(x) dx + f ′+(θ0)

2π

∫ π

0xkn(x) dx

= f (θ0) + [f ′+(θ0) − f ′

−(θ0)] 1

2π

∫ π

0xkn(x) dx +

+ 1

4πn

{∫ 0

−π

R−(x, θ0;f )sin2(x/2)

dx +∫ π

0

R+(x, θ0;f )sin2(x/2)

dx

}−

− 1

4πn

{∫ 0

−π

R−(x, θ0;f )sin2(x/2)

cos(nx) dx +

+∫ π

0

R+(x, θ0;f )sin2(x/2)

cos(nx) dx

}.

The result follows from Lemma 1 and the Riemann–Lebesgue Lemma. ✷The result of Theorem 7 is valid when f ′(θ) exists. Its leading term is consistent

with Theorem 8, but the subsequent terms differ. Our goal is to obtain a uniform


expression. Let us now assume that θ is different from θ0, where θ0 is a discontinu-ity of f ′. We will again rely on the Riemann–Lebesgue Lemma, of which we notethe following refinement:

LEMMA 2. If φ is an integrable function on [−π, π ], then∫ b

acos(nx)φ(x) dx →

0 as n → ∞, uniformly in a and b for −π � a � b � π .Proof. It is easy to see that if χ[a,b] denotes the characteristic function of [a, b]

then (a, b) → χ[a,b]φ is a continuous map into L1([−π, π ]), so the set of suchfunctions is compact, and the uniformity now follows from the remark to Theo-rem 2.8 on p. 13 of [12]. ✷

Let

K = 1

4π

∫ π

−π

x2

sin2(x/2)dx = 2.77259 . . . . (36)

LEMMA 3. Assume θ �= 0. We have, uniformly in θ:

1

2π

∫ π

θ

kn(x) dx

={

12 − 1

nπ

(cosnθ

θ+ n Si(nθ) − 1

2 cot(θ2

)) + O(n−2), if θ > 0,12 + 1

nπ

(cosnθ

θ+ n Si(nθ) − 1

2 cot(θ2

)) + O(n−2), if θ < 0,(37)

1

2π

∫ π

θ

xkn(x) dx

= 1

πn

(Ci(n|θ |) + |θ |

2cot

( |θ |2

)− log sin

( |θ |2

))+ O(n−2), (38)

and1

2π

∫ π

−π

x2kn(x) dx = K

n+ O(n−2). (39)

Proof. Assume that θ > 0. By (27) we have

1

2π

∫ π

θ

kn(x) dx = 1

4πn

∫ π

θ

[1

sin2(x/2)− cos(nx)

(x/2)2+

+ cos(nx)

(1

(x/2)2− 1

sin2(x/2)

)]dx

= 1

2πncot

(θ

2

)− 1

π

∫ nπ

nθ

cos(t)

t2dt + O(n−2)

= 1

2πncot

(θ

2

)− 1

π

[−cos(nπ)

nπ+ cos(nθ)

nθ−

− Si(nπ) + Si(nθ)

]+ O(n−2).


Similarly (38) is an even function of θ since xkn(x) is an odd function of x, sowe may assume θ > 0 in evaluating it. Then

1

2π

∫ π

θ

xkn(x) dx

= 1

4πn

∫ π

θ

x(1 − cos(nx))

[4

x2+ 1

sin2(x/2)− 1

(x/2)2

]dx

= 1

πn(log(π) − log(θ) − Ci(nπ) + Ci(nθ)) +

+ 1

4πn

∫ π

θ

x

(1

sin2(x/2)− 1

(x/2)2

)dx −

− 1

4πn

∫ π

θ

x

(1

sin2(x/2)− 1

(x/2)2

)cos(nx) dx

= 1

πn

(Ci(nθ) − Ci(nπ) + θ

2cot

(θ

2

)− log sin

(θ

2

))−

− 1

4πn

∫ π

θ

x

(1

sin2(x/2)− 1

(x/2)2

)cos(nx) dx

= 1

πn

(Ci(nθ) + θ

2cot

(θ

2

)− log sin

(θ

2

))+ O(n−2).

Finally, the difference between the left and right sides of (39) is

1

4πn

∫ π

−π

x2

sin2(x/2)cos(nx) dx = O(n−2)

since the integrand is regular at its endpoints. ✷We will make use of the following function:

H(x) ={

12 (x + π)2, −2π � x � 0;12 (x − π)2, 0 � x � 2π.

We note that this function satisfies H(x + 2π) = H(x) when x and x + 2π areboth in its range, so H has an extension to a 2π periodic function. Its derivativehas a discontinuity at 0, which is unique modulo 2π .

LEMMA 4. We have, recalling �n(θ) from (10),

(H ∗ kn)(θ) = H(θ) + K

2n− 1

n�n(θ) + O(n−2).


Proof. We have, using the fact that kn is even, and Lemma 3:

(H ∗ kn)(θ) = 1

4π

∫ −θ

−π

(x + θ + π)2kn(x) dx +

+ 1

4π

∫ π

−θ

(x + θ − π)2kn(x) dx

= 1

4π

∫ π

−π

x2kn(x) dx −∫ π

θ

xkn(x) dx +

+ 12 (θ

2 + π2)

[1

2π

∫ π

θ

kn(x) dx + 1

2π

∫ π

−θ

kn(x) dx

]−

−πθ

[1

2π

∫ π

θ

kn(x) dx + 1

2π

∫ π

−θ

kn(x) dx

]

= K

2n+ 1

2(θ2 + π2) − 2

n

(Ci(nθ) − log sin

(θ

2

)+

+ cos(nθ) + nθ Si(nθ)

)+ O(n−2).

Adding πθ to the second term and subtracting it from the third gives

K

2n+ H(θ) − 1

n�n(θ) + O(n−2). ✷

THEOREM 9. Suppose that f has jump discontinuities in its derivative at θi , andthat these are the only discontinuities in f ′ modulo 2π . Let

αi = 1

2π(f ′

+(θi) − f ′−(θi)).

Assume that φθ(x) = x−2R(x, θ;f ) is integrable on (−π, π), where R(x, θ;f ) isdefined by (31), or by (34) if θ is a θi . Then with �n as in (10), we have

(f ∗ kn)(θ) = f (θ) +∑i

αin−1�n(θ − θi) +

+ 1

4πn

∫ π

−π


dx + o(n−1). (40)

If θ �→ φθ is a continuous map of [−π, π ] to L1([−π, π ]), then (40) is uniformin θ . If φ′

θ exists and is integrable, the error in (40) is O(n−2).

In applying this theorem, note that as we move θ around the interval, we wantto keep |θi − θ | � π . This means that representatives θi are chosen differently


depending on the location of θ . We’ve done this implicitly by defining �n(θ)

by (10) when |θ | � π , and extending it to a 2π periodic function.

Proof. Let f0 = f+∑i αiHi , where Hi(x) = H(x−θi). The function f0 is con-

tinuous and has a continuous first derivative, so we may apply Theorem 7. We have

(f0 ∗ kn)(θ) − f0(θ) = 1

4πn

∫ π

−π

R(x, θ;f0)

sin2(x/2)dx + o(n−1). (41)

If φ′θ (x) exists is integrable, then Theorem 7 further asserts that the error is O(n−2).

This estimate may be shown to be uniform in θ along the lines of Lemma 2. ByLemma 4, the left side of (41) equals

1

2π

∫ π

−π

f (x + θ)kn(x) dx − f (θ) +

+∑i

αiK

2n−

∑i

αin−1�n(θ − θi) + O(n−2).

We check easily that for the quadratic functions Hi we have R(x, θ;Hi) = 12x

2

independent of θ and θi , so using (36) we have

1

4πn

∫ π

−π

R(x, θ;f0)

sin2(x/2)dx = 1

4πn

∫ π

−π


dx +∑i

αiK

2n. (42)

Comparing, we obtain (40). ✷In an important special case, the result can be made more explicit.

THEOREM 10. Let f be a continuous, piecewise linear 2π periodic function, andlet θi and αi be as in Theorem 9. Then

(f ∗ kn)(θ) = f (θ) +∑i

αin−1�n(θ − θi) + O(n−2)

uniformly in θ .Proof. It is easy to see that φ′

θ exists and is integrable. The theorem will thusfollow from Theorem 9 provided we show that

1

4π

∫ π

−π


dx = 0. (43)

Since f ′+(θi) = f ′−(θi+1), we have∑

i αi = 0. Furthermore, on each interval(θi, θi+1) the function f0 defined in the proof of Theorem 9 is polynomial ofdegree � 2, and the coefficient of x2 is 1

2

∑i αi = 0, so f0 is piecewise lin-

ear with continuous derivative and 2π periodic; therefore f0 is constant. ThusR(x, θ;f0) = 0, and (43) now follows from (42). ✷

Let λ be a partition, and let kn,λ(θ) = Kn,λ(eiθ ).


THEOREM 11. There exists a function �n,λ(θ) such that if f is a piecewise linearand continuous 2π periodic function, then with αi as in Theorems 9 and 10, wehave

(f ∗ kn,λ)(θ) = f (θ) +∑i

αin−1�n,λ(θ − θi) + O(n−2).

The function �n,λ − �n is a trigonometric polynomial, and is independent of n.Proof. Let f0 = f + ∑

i αiHi , where Hi(θ) = H(θ − θi) as in the proof ofTheorems 9 and 10. It was shown in the proof of Theorem 10 that f0 is constant.By Theorem 10, we have

(f ∗ kn,λ − f )(θ)

=∑

αin−1�n(θ − θi) +

(f0 −

∑αiHi

)∗ (kn,λ − kn)(θ) + O(n−2)

=∑i

αin−1�n(θ − θi) −

∑i

αiHi ∗ (kn,λ − kn)(θ) + O(n−2), (44)

since f0 is constant and kn,λ − kn has mean value 0. By Theorem 3(ii), there existpolynomials fλ and gλ in t and t−1 such that

(kn,λ − kn)(θ) = 1

n(fλ(e

iθ ) + einθ gλ(eiθ ) + e−inθ gλ(e

−iθ )).

Moreover, f (t) = f (t−1), so the latter expression may be written as an eventrigonometric polynomial – a finite linear combination of functions cosk(θ) =cos(kθ). Substituting this into the right-hand side of (44), the convolution maybe worked out using

(Hi ∗ cosk)(θ) = 1

k2cosk(θ − θi).

Thus fλ contributes n−1 ∑αiG(θ − θi), where G is a fixed trigonometric poly-

nomial, while gλ contributes terms of order O(n−2) which may be discarded. Thetheorem follows. ✷

References

1. Bump, D. and Diaconis, P.: Toeplitz minors, J. Combin. Theory Ser. A 97 (2002), 252–271.2. Coram, M. and Diaconis, P.: New tests of the correspondence between unitary eigenvalues and

the zeros of Riemann’s zeta function, to appear in Ann. Statist.3. Diaconis, P. and Evans, S.: Linear functionals of eigenvalues of random matrices, Trans. Amer.

Math. Soc. 353 (2001), 2615–2633.4. Diaconis, P. and Shahshahani, M.: On the eigenvalues of random matrices, In: Studies in

Applied Probability, a special volume of J. Appl. Probab. A 31 (1994), 49–62.


5. Dyson, F.: Statistical theory of the energy levels of complex systems, I, II, III, J. Math. Phys. 3(1962), 140–156, 157–165, 166–175.

6. Dyson, F.: Correlations between eigenvalues of a random matrix, Comm. Math. Phys. 19(1970), 235–250.

7. Fejér, L.: Untersuchungen über Fouriershe Reihen, Math. Ann. 58 (1904), 501–569.8. Goodman, R. and Wallach, N.: Representations and Invariants of the Classical Groups,

Cambridge Univ. Press, 1998.9. Hejhal, D., Friedman, J., Gutzwiller, M. and Odlyzko, A. (eds): Emerging Applications of

Number Theory, Springer-Verlag, New York, 1999.10. Hughes, C.: On the characteristic polynomial of a random unitary matrix and the Riemann zeta

function, Dissertation, University of Bristol, 2001.11. Hughes, C., Keating, J. and O’Connell, N.: On the characteristic polynomial of a random

unitary matrix, Comm. Math. Phys. 220 (2001), 429–451.12. Katznelson, Y.: An Introduction to Harmonic Analysis, 2nd edn, Dover, New York, 1976.13. Keating, J. and Snaith, N.: Random matrix theory and ζ( 1

2 + it), Comm. Math. Phys. 214(2000), 57–89.

14. Keating, J. and Snaith, N.: Random matrix theory and L-functions at s = 12 , Comm. Math.

Phys. 214 (2000), 91–110.15. Körner, T.: Fourier Analysis, Cambridge Univ. Press, 1988.16. Macdonald, I.: Symmetric Functions and Hall Polynomials, 2nd edn, Oxford Univ. Press, 1995.17. Mehta, M.: Random Matrices, 2nd edn, Academic Press, New York, 1991.18. Pinsky, M.: Fejér asymptotics and the Hilbert transform, Preprint, Department of Math., North-

western Univ., 17 April 2001. To appear in Amer. Math. Soc. Contemp. Math. Ser. (A. Seegeret al. (eds)).

19. Rains, E.: High powers of random elements of compact Lie groups, Probab. Theory RelatedFields 107(2) (1997), 219–241.

20. Soshnikov, A.: Level spacings distribution for large random matrices: Gaussian fluctuations,Ann. of Math. (2) 148 (1998), 573–617.

21. Soshnikov, A.: Central limit theorem for local linear statistics in classical compact groups andrelated combinatorial identities, Ann. Probab. 28 (2000), 1353–1370.

22. Stanley, R.: Enumerative Combinatorics, Cambridge Univ. Press, 1986, 1997, 1999.23. Taylor, M.: Multi-dimensional Fejér kernel asymptotics, Preprint, Dept. of Math., Univ. North

Carolina, Chapel Hill, 2001.24. Tracy, C. and Widom, H.: Introduction to random matrices, In: Geometric and Quantum As-

pects of Integrable Systems (Scheveningen, 1992), Lecture Notes in Phys. 424, Springer-Verlag,New York, 1993, pp. 103–130.

25. Tracy, C. and Widom, H.: Correlation functions, cluster functions, and spacing distributions forrandom matrices, J. Statist. Phys. 92 (1998), 809–835.

26. Wieand, K.: Eigenvalue distributions of random matrices in the permutation group and compactLie groups, Harvard PhD Dissertation, 1998.

27. Zygmund, A.: Trigonometric Series, 2nd edn, Cambridge Univ. Press, 1959.


125

Trajectories Joining Two Submanifolds under theAction of Gravitational and ElectromagneticFields on Static Spacetimes

ROSSELLA BARTOLO1,� and ANNA GERMINARIO2,��1Dipartimento Interuniversitario di Matematica, Politecnico di Bari, Via E. Orabona, 4,70125 Bari, Italy2Dipartimento Interuniversitario di Matematica, Università degli Studi di Bari, Via E. Orabona, 4,70125 Bari, Italy

(Received: 27 September 2001; accepted in final form: 14 March 2002)

Abstract. In this paper we present existence and multiplicity results for orthogonal trajectoriesjoining two submanifolds under the action of gravitational and electromagnetic fields on static space-times. These trajectories are critical points of unbounded functionals and they can be found by usinga variant of the saddle point theorem and the relative category theory.

Mathematics Subject Classifications (2000): 58E10, 58E05, 53C50.

Key words: Lorentzian manifold, normal trajectory, saddle-points, relative category.

1. Introduction

A pair (L, g) is called a Lorentzian manifold if L is a connected finite-dimensionalsmooth manifold with dimL � 2 and g is a Lorentzian metric on L, that is g is asmooth, symmetric, two covariant tensor field such that, for any z ∈ L, the bilinearform g(z)[·, ·] induced on TzL is nondegenerate and of index ν(g) = 1. The pointsof L are called events. A Lorentzian manifold (L, g) is called (standard) stationaryif L is a product manifold

L = M × R, M any C3-connected manifold

and g can be written as

〈ζ, ζ ′〉L = 〈ξ, ξ ′〉 + 〈δ(x), ξ 〉τ ′ + 〈δ(x), ξ ′〉τ − β(x)ττ ′ (1)

for any

z = (x, t) ∈ L, ζ = (ξ, τ ), ζ ′ = (ξ ′, τ ′) ∈ TzL = TxM × R,

� Work supported by MURST (ex 40% and 60% research funds).�� Work supported by MURST (ex 40% and 60% research funds).

126 ROSSELLA BARTOLO AND ANNA GERMINARIO

where 〈·, ·〉, δ and β are, respectively, a Riemannian metric onM, a smooth vectorfield and a smooth scalar field on M. When δ ≡ 0, L is called (standard) static.We refer to [21] and [23] for all the background material assumed in this paper.

Let us consider a smooth stationary vector field A on L, that is ∂tA1(z) =∂tA2(z) = 0, thus

A(z) = A(x, t) = A(x) = (A1(x),A2(x)), ∀z = (x, t) ∈ L.In some recent papers, the existence and the multiplicity of trajectories (underthe action of A) joining two events in L has been studied. Namely, fixed twoevents z,w ∈ L, the trajectories joining them satisfy the Euler–Lagrange equationassociated to the functional introduced in [6]

F(γ ) = 1

2

∫ 1

0〈γ , γ 〉L ds +

∫ 1

0〈A (γ ), γ 〉L ds (2)

on

�(z,w;L) = {γ ∈ H 1([0, 1], L) | γ (0) = z, γ (1) = w}

,

that is

Dsγ = ((A′(γ ))∗ − A′(γ ))[γ ], (3)

where A′ is the differential of the vector field A and (A′(z))∗ denotes, for anyz ∈ L, the adjoint operator of A′(z) on TzL with respect to 〈·, ·〉L.

This problem has been studied in [2] and [12] on complete stationary Lorentzianmanifolds, in [3] on open subsets of stationary Lorentzian manifolds, and in [1]in a more general setting. It is clear that this problem generalizes the geodesicconnectedness one (see, e.g., [7, 16]).

In this paper we extend the results in [2] and [12]. Indeed, we shall look fororthogonal trajectories under a vectorial potential joining two given submanifoldsof a stationary Lorentzian manifold L.

DEFINITION 1.1. Let S,� be two submanifolds of L. A curve γ : [0, 1] → L iscalled orthogonal trajectory (under the action of A) joining S to � if

(i) γ satisfies (3),

(ii)

{γ (0) ∈ S, γ (1) ∈ �,γ (0) ∈ Tγ (0)S⊥, γ (1) ∈ Tγ (1)�⊥.

This problem has been studied when A ≡ 0 in [19] and [9, 10], respectively,on stationary and on orthogonal splitting Lorentzian manifolds. For the sake ofsimplicity, we shall deal with static Lorentzian manifolds, although our results holdalso for stationary Lorentzian manifolds under some additional assumptions on thecoefficient δ (see (1)).

TRAJECTORIES JOINING TWO SUBMANIFOLDS UNDER THE ACTION 127

Let P and Q be two submanifolds ofM and let us set

S1 = P × {0}�, S2 = Q× {T }, T ∈ R, S3 = Q× R. (4)

We shall prove existence and multiplicity results for orthogonal trajectoriesjoining, respectively, S1 to S2 and S1 to S3.

It can be easily proved (see Proposition 2.1) that, if A is orthogonal to Si , i =1, 2, 3, that is

〈A(z), ζ 〉L = 0, ∀z ∈ S1 ∪ Si, ζ ∈ Tz(S1 ∪ Si), i = 2, 3, (5)

then the orthogonal trajectories joining S1 to Si (i = 2, 3) are the critical points ofF at (2) on a suitable Hilbert manifold (see Section 2).

Before stating our main results, we recall that a vector ζ ∈ TzL is called timelike(respectively lightlike; spacelike) if 〈ζ, ζ 〉L < 0 (respectively, 〈ζ, ζ 〉L = 0, ζ �= 0;〈ζ, ζ 〉L > 0 or ζ = 0). We remark that Equation (3) has a prime integral, in fact

d

ds〈γ , γ 〉L = 2〈Dsγ , γ 〉L = 〈(A′(γ ))∗[γ ] − A′(γ )[γ ], γ 〉L = 0,

hence if γ : [0, 1] → L is a trajectory, there exists Eγ ∈ R such that

〈γ (s), γ (s)〉L = Eγ , ∀s ∈ [0, 1]. (6)

Therefore a trajectory γ is said to be timelike, lightlike or spacelike according tothe causal character of γ .

Let us assume that there exist η, b ∈ R such that

0 < η � β(x) � b, ∀x ∈ M; (7)

there exist a1, a2 ∈ R such that

|A1(x)| � a1 and 0 � A2(x) � a2, ∀x ∈ M; (8)

P and Q are closed submanifolds ofM such that P orQ is compact; (9)

P and Q are disjoint. (10)

Remark 1.2. A Gauge transformation does not modify the set of the criticalpoints of the functional F . Indeed adding to A any irrotational vector field Yindependent of t , say Y = (∇V (x), a) with V ∈ C2(M,R) and a ∈ R, thecritical points of the corresponding functional still satisfy (3) if (5) holds. Thus,it is enough that A + Y satisfies (8) for such Y (in particular, it suffices that A2 isbounded from below).

The following theorems concern, respectively, the existence and the multiplicityof normal trajectories joining S1 to S2 and they will be proved in Section 3.� We could consider S1 = P × {t0}, t0 ∈ R, however, as the metric is stationary, there is not loss

of generality if we assume t0 = 0.


THEOREM 1.3. LetL = M×R be a static Lorentzian manifold withM complete.Assume that (5), (7), (8), (9), (10) hold. Then an orthogonal trajectory joining S1

to S2 exists. Moreover, T > 0 exists such that for any T ∈ R with |T | > T there isan orthogonal timelike trajectory joining S1 to S2.

THEOREM 1.4. Let the assumptions of Theorem 1.3 hold. IfM is not contractiblein itself and P ,Q are both contractible inM, then

(i) there exists a sequence {γm} of (spacelike) trajectories joining S1 to S2 suchthat limm→+∞Eγm = +∞;

(ii) denoted by N(T, S1, S2) the number of the timelike orthogonal trajectoriesjoining S1 to S2, it results lim|T |→+∞ N(T, S1, S2) = +∞.

The previous results about spacelike trajectories in Theorems 1.3 and 1.4 haveonly a geometrical meaning, while the results concerning timelike trajectories havealso a physical interpretation. Indeed, the Lorentz world-force law which determi-nates the motion of relativistic particles γ submitted to an electromagnetic field, isthe Euler–Lagrange equation related to the action functional

S(γ ) = −m0c1

2

∫ s1

s0

√−〈γ , γ 〉L ds + q∫ s1

s0

〈A (γ ), γ 〉L ds,

where m0 is the rest mass of the particle, q is its charge, c is the speed of light (see[23, p. 88]). In [6], it is proved that for timelike trajectories the search of criticalpoints of S is equivalent to that of the critical points of F . In particular, for Eγ < 0,the inertial mass turns out to be a constant of the motion, which is determined bythe initial conditions and also the equality between the inertial and gravitationalmass can be deduced (see [6]).

Remark 1.5. We point out that if L is a static Lorentzian manifold and wereplace (7) in Theorems 1.3 and 1.4 by there exist b ∈ R such that 0 < β(x) �b ∀x ∈ M, we still get the existence of a trajectory, but we are not able to findtimelike trajectories.

The following theorems concern, respectively, the existence and the multiplicityof normal trajectories joining S1 to S3 and they will be proved in Section 4.

THEOREM 1.6. Let the assumptions of Theorem 1.3 hold. Then there exists a(spacelike) orthogonal trajectory joining S1 to S3.

THEOREM 1.7. Let the assumptions of Theorem 1.4 hold. Then there exists asequence {γm} of (spacelike) trajectories joining S1 to S3.

Remark 1.8. We point out that if γ = (x, t) is a normal trajectory joining S1

to S3, as Tγ (1)S3 = Tx(1)Q× R, necessarily t (1) = 0, then, from (6) and (1), γ hasto be spacelike.


Remark 1.9. Evaluating the Fréchet differential of F it is clear that if P and Qare not disjoint and (5) holds, then for any x ∈ P ∩ Q the curve γ = (x, 0) isa trivial (lightlike) trajectory. Thus, assumption (10) is needed in Theorem 1.3 ifT = 0 in order to prove the existence of a nontrivial normal trajectory, while, ifT �= 0, it is necessary only to prove that in some cases the normal trajectory isspacelike (see Remark 3.9). Moreover, if (10) does not hold, γ is always a trivialtrajectory joining S1 to S3. Clearly, in the multiplicity results of Theorems 1.4and 1.7 assumption (10) is not needed since there exist infinitely many nontrivialtrajectories.

We have already pointed out that normal trajectories joining S1 to Si, i = 2, 3,are the critical points of the functional F on suitable Hilbert manifolds. We remarkthat F is unbounded both from above and from below, also modulo compact pertur-bations, hence the search of its critical points is not trivial. Nevertheless, since thecoefficients of the metric (1) do not depend on the variable t , it is possible to provea variational principle (see Proposition 3.1 and [7]) which reduces our problem tothe study of a functional depending only on the ‘spatial’ component.

• If we look for normal trajectories γ = (x, t) joining S1 to S2, as t (1) = Tis fixed, the classical Ljusternik–Schnirelmann critical point theory can beapplied (see Section 3): indeed the new functional is bounded from below if(7) and (8) hold, and satisfies the well-known Palais–Smale condition (seeDefinition 3.3).

• If we look for normal trajectories γ = (x, t) joining S1 to S3, as t (1) freelyvaries in R, the new functional is still unbounded both from below and fromabove, hence we shall use a different approach. Thanks to the stationarity ofthe metric, the functional F satisfies the Palais–Smale condition. Then weshall introduce a Galerkin approximation scheme in the variable t , and, by avariant of the Rabinowitz saddle point theorem, we shall find a critical point ofF (i.e. a normal trajectory joining S1 to S3). In order to get multiplicity results,we shall use the relative category for unbounded functionals (see [13, 15, 25]).

Remark 1.10. Plainly, if P and Q reduce respectively to {p} and {q}, then weobtain the results in [2] and [12] for trajectories under a vectorial potential joiningtwo fixed events in L.

2. The Functional Setting

Hereafter we shall assume that M is a submanifold of RN for N sufficiently large(see [20]), thus

H 1([0, 1], L) = {z ∈ H 1([0, 1],RN+1) | z([0, 1]) ⊂ L}

,


where

H 1([0, 1],RN ) ≡ H 1,2([0, 1],RN)= {y ∈ L2([0, 1],RN ) | y is absolutely continuous,

y ∈ L2([0, 1],RN)}.We shall denote by ‖ · ‖ the usual norm on H 1([0, 1],RN) and by ‖ · ‖2 the usualnorm on L2([0, 1],RN).

Let us set for i = 2, 3

0(S1, Si;L) ={z ∈ H 1([0, 1], L) | z(0) ∈ S1, z(1) ∈ Si

},

then, for any z ∈ 0(S1, Si;L), i = 2, 3,

Tz0(S1, Si;L) ={ζ ∈ TzH 1([0, 1], L) | ζ(0) ∈ Tz(0)S1, ζ(1) ∈ Tz(1)Si

}.

By using standard arguments (see, e.g., [18]) we can prove the following proposi-tion:

PROPOSITION 2.1. Let γ ∈ 0(S1, Si;L), i = 2, 3 and assume that (5) holds.Then γ is a critical point of F if and only if it is an orthogonal trajectory joiningS1 and Si, i = 2, 3.

By Proposition 2.1, the orthogonal trajectories joining S1 to S2 are the criticalpoints of F on

ZT := 0(S1, S2;L) = �(P,Q;M) ×H 1(0, T ),

where

�(P,Q;M) = {x ∈ H 1([0, 1],M) | x(0) ∈ P, x(1) ∈ Q}

is a smooth submanifold of H 1([0, 1],M) (see [18]) and

H 1(0, T ) = {t ∈ H 1([0, 1],R) | t (0) = 0, t (1) = T }

.

For any z = (x, t) ∈ ZT , it results that

TzZT = Tx�(P,Q;M) ×H 10 ([0, 1],R),

where

Tx�(P,Q;M) ={ξ ∈ TxH 1([0, 1],M) | ξ(0) ∈ Tx(0)P , ξ(1) ∈ Tx(1)Q

}and

H 10 ([0, 1],R) =

{τ ∈ H 1([0, 1],R) | τ(0) = 0 = τ(1)}.


Remark 2.2. If γ = (x, t) is a trajectory joining S1 to S2, (ii) of Definition 1.1and (5) can be respectively written as

x(0) ∈ P, t (0) = 0, x(1) ∈ Q, t(1) = T ,x(0) ∈ Tx(0)P⊥, x(1) ∈ Tx(1)Q⊥,

〈A1(x), ξ 〉 = 0 ∀x ∈ P ∪Q ∀ξ ∈ TxP ∪ TxQ.On the other hand, by Proposition 2.1, the orthogonal trajectories joining S1 to S3

are the critical points of F on

Z := 0(S1, S3;L) = �(P,Q;M) ×W,where

W = {t ∈ H 1([0, 1],R) | t (0) = 0

}.

By virtue of the Poincaré inequality, the space W can be equipped with the normequivalent to ‖ · ‖

‖t‖eq = ‖t‖2. (11)

We remark that

W = H 10 ([0, 1],R) ⊕ Rj[0,1] with j[0,1]: s ∈ [0, 1] → s ∈ R.

For any z = (x, t) ∈ Z, it results that TzZ = Tx�(P,Q;M) ×W .

Remark 2.3. If γ = (x, t) is a trajectory joining S1 to S3, (ii) of Definition 1.1can be written as

x(0) ∈ P, t (0) = 0, x(1) ∈ Q,x(0) ∈ Tx(0)P⊥, x(1) ∈ Tx(1)Q⊥, t (1) = 0.

3. Proof of Theorems 1.3 and 1.4

As pointed out in Section 2, normal trajectories joining S1 to S2 are the criticalpoints of FT := F on ZT . We have already observed that, as for the geodesic prob-lem on Lorentzian manifolds (see [5, 7]), the functional FT is strongly indefinite.We can overcome such difficulty by a slight variant of the variational principlein [2] which extends the one proved in [7] and which reduces the study of theorthogonal trajectories joining S1 to S2 to the search of the critical points of asuitable functional, defined only on �(P,Q;M), which is bounded from belowunder our assumptions. Indeed, the following proposition holds.

PROPOSITION 3.1. Let γ = (x, t) ∈ ZT . The following propositions are equiv-alent:


(a) γ is a critical point of FT ;(b) (i) x ∈ �(P,Q;M) is a critical point of the C2 functional JT defined on

�(P,Q;M) by

JT (x) = 1

2

∫ 1

0〈x, x〉 ds +

∫ 1

0〈A1(x), x〉 ds +

+1

2

∫ 1

0β(x)A2

2(x) ds − 1

2

(T + ∫ 1

0 A2(x) ds)2∫ 1

01β(x)

ds, (12)

(ii) t ∈ H 1(0, T ) is the solution of the following Cauchy problem:

t = H(x)β(x)

− A2(x), t (0) = 0, (13)

where

H(x) = T +∫ 1

0 A2(x) ds∫ 10

1β(x)

ds. (14)

Moreover, if (a) or (b) is true, FT (γ ) = JT (x).Remark 3.2. From (7), (8), (12) and the Hölder inequality we get, for any x ∈

�(P,Q;M),

JT (x) � 12‖x‖2

2 − a1‖x‖2 − b(T 2

2+ a

22

2+ T a2

), (15)

hence JT is bounded from below.

In the remainder of this section we shall denote by X a C2 Hilbert manifoldendowed with a Riemannian metric. We shall prove Theorem 1.4 by using theLjusternik–Schnirelmann category theory for functionals bounded from below. Letus recall some definitions and results (see, e.g., [24]).

DEFINITION 3.3. Let f ∈ C1 (X,R); f satisfies the Palais–Smale condition ifevery sequence {ym} such that

{f (ym)} is bounded (16)

and

limm→+∞ ‖f

′(ym)‖∗ = 0 (17)

contains a converging subsequence (where ‖ · ‖∗ is the norm induced on the cotan-gent bundle by the Riemannian metric on X).

DEFINITION 3.4. Let A be a subspace of X. The category of A in X, denoted bycatX A, is the minimum number of closed and contractible subsets of X coveringA (possibly +∞). We shall write catX = catX X.


THEOREM 3.5. Let f ∈ C1(X,R) be a functional bounded from below, satisfy-ing the Palais–Smale condition and let X be complete. Then f has at least catXcritical points. Moreover, if catX = +∞, there exists a sequence {ym} of criticalpoints of f such that limm→+∞ f (ym) = +∞.

We shall obtain multiplicity results thanks to Theorem 3.5 and the followingtheorem (see [11, 14]).

THEOREM 3.6. Let M be a noncontractible in itself C3 Riemannian manifold.Let P and Q be two submanifolds of M both contractible inM. Then there existsa sequence {Km} of compact subsets of �(P,Q;M) such that

limm→+∞ cat�(P,Q;M) Km = +∞.

In order to prove the Palais–Smale condition, we recall the following lemmawhose proof is essentially contained in [5] (see also [19]).

LEMMA 3.7. Assume that P and Q are two closed submanifold of a completeRiemannian manifold M. Let {xm} be a sequence in �(P,Q;M) weakly converg-ing to a x ∈ H 1([0, 1],RN). Then x ∈ �(P,Q;M) and there exist two sequences{ξm} and {νm} in H 1([0, 1],RN) such that

xm − x = ξm + νm,ξm ∈ Txm�(P,Q;M),ξm→ 0 weakly in H 1([0, 1],RN),νm→ 0 strongly in H 1([0, 1],RN).

PROPOSITION 3.8. The functional JT (see (12)) satisfies the Palais–Smale con-dition.

Proof. Let {xm} be a Palais–Smale sequence. By Remark 3.2, we get that

{‖xm‖2} is bounded. (18)

Assumption (9) and (18) imply that {xm} is bounded in H 1([0, 1],RN). Hence,x ∈ H 1([0, 1],RN) exists such that, up to a subsequence,

xm→ x uniformly. (19)

By Lemma, 3.7, x ∈ �(P,Q;M) since P andQ are both closed inM. From (17),(12), and (14), we easily get


J ′T (xm)[ξm] =∫ 1

0〈xm, ξm〉 ds +

∫ 1

0〈A′1(xm)[ξm], xm〉 ds +

+∫ 1

0〈A1(xm), ξm〉 ds + 1

2

∫ 1

0〈∇β(xm), ξm〉A2

2(xm) ds +

+∫ 1

0β(xm)A2(xm)〈∇A2(xm), ξm〉 ds −

−H(xm)∫ 1

0〈∇A2(xm), ξm〉 ds −

−1

2H 2(xm)

∫ 1

0

〈∇β(xm), ξm〉β2(xm)

= o(1), (20)

where o(1) denotes an infinitesimal sequence and {ξm} is as in Lemma 3.7. From(18), the regularity of β,A1, A2, the uniform convergence of {ξm} to 0 and from(20) we get

o(1) =∫ 1

0〈xm, ξm〉 ds. (21)

From (21) and Lemma 3.7, we obtain

o(1) =∫ 1

0〈xm, xm − x〉 ds (22)

and since

xm→ x weakly in L2([0, 1],RN ),we have

o(1) =∫ 1

0〈x, xm − x〉 ds +

∫ 1

0〈xm − x, xm − x〉 ds, (23)

and then

xm→ x strongly in L2([0, 1],RN). (24)

As L∞([0, 1],RN) is embedded in L2([0, 1],RN), from (19) we have

xm→ x strongly in L2([0, 1],RN). (25)

From (24) and (25) we deduce that

xm→ x strongly in H 1([0, 1],RN)and the proof is complete. ✷


Proof of Theorem 1.3. By Remark 3.2 and Proposition 3.8 we get the existenceof a minimum point x of JT , that is an orthogonal trajectory joining S1 and S2 (seePropositions 3.1 and 2.1). As x is a minimum point for JT , it results

cT := JT (x) � JT (y), ∀y ∈ �(P,Q;M).Therefore, from (12), for a fixed y ∈ �(P,Q;M) it results cT � c1 − 1

2ηT2 for a

suitable c1 > 0. Thus, set γ = (x, t) (see Proposition 3.1), from (6), (13) and (14)we get

1

2Eγ = 1

2〈γ , γ 〉L = cT −

∫ 1

0〈A (γ ), γ 〉L ds

� c2 + c3T − 1

2ηT 2 + a1

∫ 1

0|x| ds (26)

for suitable c2, c3 > 0. By the Young inequality

a1‖x‖2 � 14‖x‖2

2 + 4a21, (27)

Equation (15), the Hölder inequality, and (26) we get

12Eγ � c2 + c3T − 1

2ηT2 + a1

√K1 +K2T +K3T 2

for suitable K1,K2,K3, thus the theorem is proved. ✷Remark 3.9. If (10) holds, from (15) it is easy to see that, as dist(P,Q) > 0,

if 14 dist(P,Q) − 4a2

1 > 0, then for |T | small enough the trajectory found inTheorem 1.3 is spacelike.

Proof of Theorem 1.4. By the assumption made, Theorems 3.6, 3.5 and Propo-sition 3.8, we get the existence of a sequence {xm} of critical points of JT suchthat limm→+∞ JT (xm) = +∞, thus by Proposition 3.1 we get the existence of asequence {γm} of critical points of FT such that

limm→+∞FT (γm) = +∞. (28)

From (6) we have for any m ∈ N

12Eγm = FT (γm)−

∫ 1

0〈A (γm), γm〉L ds.

Standard calculations show that

12Eγm � FT (γm)− a1‖xm‖2 − ba2

[b

η(T + a2)+ a2

],


hence from (27) and (15) it follows that

12Eγm �FT (γm)− a1

√FT (γm)+ 16a2

1 + 4b

(T 2

2+ a

22

2+ T a2

)−

−ba2

[b

η(T + a2)+ a2

]and from (28), (i) of Theorem 1.4 is proved. Next we shall prove that for anym ∈ N there exists T (m) > 0 such that for any T ∈ R, |T | > T (m), it results thatN(T, S1, S2) � m. By Theorem 3.6 for any fixed m ∈ N there exists a compactsubset Km of �(P,Q;M) such that cat�(P,Q;M) Km � m. Set

cp = infB∈0p

supx∈BJT (x), p = 1, . . . , m,

where

0p ={B ⊂ �(P,Q;M) | cat

�(P,Q;M)B � p

}(clearly 0p is not empty for any p = 1, . . . , m). Remark that the numbers cp, p =1, . . . , m, are well defined, therefore there exist at least m critical points of JTcorresponding to m critical points γ1, . . . , γm of FT with critical values cp, p =1, . . . , m. For any p = 1, . . . , m it results

1

2Eγp = cp −

∫ 1

0〈A (γp), γp〉L ds

and, as Km is compact,

cp = J (xp) � supx∈Km

JT (x) � c1 − 12ηT

2

for a suitable c1 > 0. Then, reasoning as in the proof of Theorem 1.3, we easily getthat Eγp < 0 for |T | large enough. ✷4. Proof of Theorems 1.6 and 1.7

We have already pointed out that the normal trajectories joining S1 to S3 are thecritical points of F on Z (see Section 2). As the metric 〈·, ·〉L does not dependon the time component, the functional F verifies the well-known Palais–Smalecondition on Z. Indeed the following proposition holds.

PROPOSITION 4.1. F satisfies the Palais–Smale condition on Z.Proof. Let {γm = (xm, tm)} ⊂ Z be a sequence satisfying (16) and (17). Set

τm = tm ∈ W ≡ TtmW . From (17) we get, in particular, the existence of aninfinitesimal sequence {εm} such that

εm‖tm‖2=F ′(γm)[(0, tm)]=−

∫ 1

0β(xm)t

2m ds −

∫ 1

0β(xm)A2(xm)tm ds.


Thus from (16) and (8) we get

εm‖tm‖2 + 12‖xm‖2

2 − a1‖xm‖2 � c

for a suitable c ∈ R. Thus

{‖tm‖2} is bounded (29)

and

{‖xm‖2} is bounded. (30)

From (9) and (30) it follows that {‖xm‖} is bounded. Hence, from (11) and (29), itfollows that {γm = (xm, tm)} is bounded in Z, thus there exists γ = (x, t) ∈ Z (infact P and Q are closed) such that

γm→ γ weakly in H 1([0, 1],RN+1). (31)

Let us show that

γm→ γ strongly in H 1([0, 1],RN+1).

Set τm = t − tm ∈ W , we have that

τm→ 0 weakly in H 1([0, 1],R). (32)

From (17)

F ′(γm)[(ξm, τm)] = o(1), (33)

where ξm is as in Lemma 3.7 and o(1) denotes an infinitesimal sequence. From(29) and (30), the regularity of β,A1, A2 and the uniform convergence of {ξm} and{τm} to 0, we get from (33)∫ 1

0〈xm, ξm〉 ds −

∫ 1

0β(xm)tmτm ds+

+∫ 1

0〈A1(xm), ξm〉 ds −

∫ 1

0β(xm)A2(xm)τm ds = o(1).

Then, as tm = t − τm, from (32) we get∫ 1

0〈xm, ξm〉 ds +

∫ 1

0β(xm)τ

2m ds = o(1),

therefore∫ 1

0〈xm, ξm〉 ds = o(1), (34)∫ 1

0β(xm)τ

2m ds = o(1). (35)


From (34) we can reason as in the proof of Proposition 3.8 obtaining

xm→ x strongly in H 1([0, 1],RN )and from (35), (32) we have

τm→ 0 strongly in H 1([0, 1],R)and the proof is complete. ✷

Due to the indefiniteness of the metric 〈·, ·〉L the functional F is unbounded oninfinite-dimensional linear manifolds. Following [5], we shall introduce a finite-dimensional Galerkin approximation scheme.

Let us set, for any k ∈ N,

Wk = Hk ⊕ Rj[0,1], where Hk = span{sinπps | p = 1, . . . , k}(see also Section 2). Our next aim is to find, for any k ∈ N, a critical point of Frestricted to

Zk = �(P,Q;M) ×Wk.Remark that by the same proof of Proposition 4.1, for any k ∈ N Fk := F|Zksatisfies the Palais–Smale condition. We shall use the following variant of the wellknown saddle point theorem (see [4, 22]).

THEOREM 4.2. LetX be a complete Riemannian manifold,H a separable Hilbertspace, Y a linear subspace of H with orthonormal basis {ym}, h ∈ H . Set

W = Y + h, Z = X ×W,Wk = span{yp | p = 1, . . . , k} + h and Zk = X ×Wk

for any k ∈ N,

S = {(x, y + h) ∈ Z | x ∈ X}, y ∈ Y,Q(R) = {(x, w) ∈ Z | ‖w − h− y‖ � R}, x ∈ X,R > 0.

Let f ∈ C1(Z,R), assume that fk := f|Zk satisfies the Palais–Smale conditionand that there exists R > 0 such that

sup f (Q(R)) < +∞, sup f (∂Q(R)) < inf f (S).

Then, for any k ∈ N there exists a critical point of fk corresponding to a criticallevel ck such that

inf f (S) � ck � sup f (Q(R)),

ck = infg∈0k

supx∈Qk(R)

fk(g(x)),

where

0k = {g ∈ C(Zk, Zk) | g(x) = x ∀x ∈ ∂Qk(R)},Qk(R) = {(x, w) ∈ Zk | ‖w − h− y‖ � R}.


PROPOSITION 4.3. For any k ∈ N there exists a critical point of Fk = F|Zk .Proof. Let us set

S = {(x, j[0,1]) ∈ Z | x ∈ �(P,Q;M)}, (36)

Q(R) = {(y, t) ∈ Z | ‖t − j[0,1]‖eq � R}, R > 0

(see (11)), where y ∈ �(P,Q;M) is fixed. From (7) and (8) it results that

F(z) � − 12b − ba2 − 1

2a21, ∀z = (x, j[0,1]) ∈ S. (37)

Again from (7) and (8), for any z = (y, t) ∈ Q(R), we easily get

F(z) � c − 12η‖t‖2

eq + ba2‖t‖eq (38)

for a suitable c, therefore supF(Q(R)) < +∞. We remark that

|R − 1| � ‖t‖eq � R + 1, ∀z = (y, t) ∈ ∂Q(R),then from (38) we have for suitable c1, c2 > 0

F(z) � c1 − 12ηR

2 + c2R, ∀z ∈ ∂Q(R).From (37), (38) and for R large enough it is supF(∂Q(R)) < infF(S). Hence byTheorem 4.2 we get, for any k ∈ N, the existence of a critical point γk of Fk suchthat

infF(S) � Fk(γk) � supF(Q(R)). � (39)

Proof of Theorem 1.6. By Proposition 4.3, we have the existence of a sequence{γk} ⊂ Z such that for any k ∈ N γk is a critical point of Fk and such that (39) holds.We shall prove that {γk} contains a subsequence converging in Z to a critical pointγ of F , concluding the proof. Indeed, reasoning as in the proof of Proposition 4.1,we get the boundedness of the sequence {γk}, therefore there exists γ = (x, t) ∈ Zsuch that, up to a subsequence, γk → γ weakly in Z. Let Pk denote the orthogonalprojection operator ofW onto Wk for any k ∈ N. Set

τk = Pk(t)− tk, for any k ∈ N.

Then

τk → 0, weakly in H 1([0, 1],R)and

Pk(t)→ t, strongly in H 1([0, 1],R).Again, reasoning as in the proof of Proposition 4.1, we have that γk → γ stronglyin Z. The same arguments used in [5, Lemma 3.4] show that γ is a critical pointof F . ✷

We recall the notion of relative category (see [13, 15, 25]).


DEFINITION 4.4. Let Y,W be two closed subsets of a topological space X.The category of W in X relative to Y , denoted by catX,Y W , is the minimumnumber m (possibily +∞) such that there exist m + 1 closed and contractiblesubsets W0, . . . ,Wm, covering W and m + 1 functions fi ∈ C([0, 1] × Wi,X),i = 0, . . . , m, such that

fi(0, w) = w, ∀w ∈ Wi, i = 0, . . . , m,

f0(1, w) ∈ Y, w ∈ W0,

f0(s, y) ∈ Y, y ∈ W0 ∩ Y, s ∈ [0, 1],fi(1, w) = wi, w ∈ Wi, for some wi ∈ X, i = 1, . . . , m.

In order to prove Theorem 1.7 we shall use the following multiplicity result (see[8] for the proof).

THEOREM 4.5. Let X be a C2 complete Riemannian manifold modelled on aHilbert space and f ∈ C1(X,R). Assume that there exist two subsets C and D ofX such that C is a closed strong deformation retract of X \D and that

infz∈Df (z) > sup

z∈Cf (z), cat

X,CX > 0.

Assume that f satisfies the Palais–Smale condition. Then f has at least catX,C Xcritical points inX with critical levels c � inf f (D). Moreover, if catX,C X = +∞,then there exists a sequence {zm} of critical points of f such that

limm→+∞ f (zm) = sup

z∈Xf (z).

In order to apply Theorem 4.5 we need the following lemma.

LEMMA 4.6. Let η, b as in (7) and a1, a2 as in (8). Then, there exists h ∈C(R+,R+) such that for any z = (x, t) ∈ Zk satisfying ‖t − j[0,1]‖eq = h(‖x‖2) itresults F(z) � −b − 2ba2 − a2

1 .Proof. For any z = (x, t) ∈ Zk it results that

F(z) � ‖x‖22 − 1

2η‖t‖2eq + 1

2a21 + ba2‖t‖eq ,

therefore, as

|‖t − j[0,1]‖eq − 1| � ‖t‖eq � ‖t − j[0,1]‖eq + 1, ∀t ∈ W,we get

F(z) � ‖x‖22 − 1

2η +a2

1

2+ ba2 + (ba2 + η)‖t − j[0,1]‖eq − 1

2η‖t − j[0,1]‖2eq . (40)

Setting

h(s) = ba2

η+ 1+

√b2a2

2

η2+ 8ba2 + 2s2 + 3a2

1 + 2b

η, s � 0,

the lemma is proved. ✷


Let us set, for any k ∈ N,

Ck = {z = (x, t) ∈ Zk | ‖t − j[0,1]‖eq = h(‖x‖2)}. (41)

Clearly, by Lemma 4.6 and (37), it follows that

supFk(Ck) < infFk(S), (42)

where S is as in (36).

Remark 4.7. For any k ∈ N the following results hold (see [10, Lemmas 7.3,7.4] for the proof).

(i) the set Ck at (41) is a closed strong deformation retract of Zk \ S;(ii) ifM is a 1-connected Riemannian manifold then for any m ∈ N there exists a

compact subset Km of Zk such that catZk,Ck Km � m.

In the proof of Theorem 1.7 we can assume thatM is a 1-connected Riemannianmanifold. Indeed, if the fundamental group π1(M) is not trivial and finite, (ii) holds(it suffices to consider the universal covering of M). On the other hand, if π1(M)

is not finite we can find a critical point of F on each connected component.

Proof of Theorem 1.7. By virtue of Remark 4.7 and (42), Theorem 4.5 can beapplied to each Fk , k ∈ N. Theorefore, for any k ∈ N we get the existence of asequence {γ km} of critical points of Fk such that

infFk(S) � Fk(γ km), ∀m ∈ N,

and, as in our case catZk,Ck Zk = +∞ (see, e.g., [10, Theorem 3.10]),limm→+∞ Fk(γ km) = +∞.

We remark that the critical levels in Theorem 4.5 are characterized by

Fk(γkm) = inf

B∈0kmsupz∈BFk(z), ∀m ∈ N,

where

0km ={B ∈ Zk | B is closed, cat

Zk,CkB � m

}.

Fix m ∈ N; for any k ∈ N, z ∈ Zk it results that

Fk(z) � ‖x‖22 + 1

2a21 + ba2 + (ba2 + η)h(‖x‖2)

(see (40)) hence, reasoning as in [10, Lemma 7.4] it results for a suitable cm > 0:

Fk(γkm) � cm, ∀k ∈ N. (43)

Now fix c > 0. There exists m = m(c) ∈ N, independent of k ∈ N, such that forany m ∈ N,m � m and B ∈ 0km

B ∩ (Ac × {j[0,1]}) �= ∅, (44)


where

Ac ={x ∈ �(P,Q;M) | 1

2

∫ 1

0〈x, x〉 ds � c

}.

Indeed, if B ∈ 0km is such that B∩(Ac×{j[0,1]}) is empty, then by (i) of Remark 4.7,it can be proved that

cat�(P,Q;M)

Ac � catZk,Ck

Km � m, (45)

where

Ac ={x ∈ �(P,Q;M) | 1

2

∫ 1

0〈x, x〉 ds � c

}

(see [10] for the details). Remark that, as M is complete and (9) holds, the Rie-mannian action functional satisfies the Palais–Smale condition (see, e.g., [17]).Therefore cat�(P,Q;M) Ac is finite and from (45), for m large enough (44) holds.From (44) and (37) it follows that

Fk(γkm) � c − 1

2 (b − 2ba2 − a21), ∀m � m. (46)

From (43) and (46), reasoning as in the proof of Theorem 1.6, we get, for anym � m, the existence of a critical point γm of F satisfying

c − 12(b − 2ba2 − a2

1) � F(γm) � cm.

As c is arbitrary, we get

limm→+∞F(γm) = +∞. �

References

1. Antonacci, F., Giannoni, F. and Magrone, P.: On the problem of the existence for connectingtrajectories under the action of gravitational and electromagnetic fields, Differential Geom.Appl. 13 (2000), 1–17.

2. Bartolo, R: Trajectories connecting two events of a Lorentzian manifold in the presence of avector field, J. Differential Equations 153 (1999), 82–95.

3. Bartolo, R.: Trajectories under a vectorial potential on stationary manifolds, Internat. J. Math.Math. Sci., to appear.

4. Benci, V. and Fortunato, D.: Existence of geodesics for the Lorentz metric of a stationarygravitational field, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), 27–35.

5. Benci, V. and Fortunato, D.: On the existence of infinitely many geodesics on space-timemanifolds, Adv. Math. 105 (1994), 1–25.

6. Benci, V. and Fortunato, D.: A new variational principle for the fundamental equations ofclassical physics, Found. Phys. 28(2) (1998), 333–352.

7. Benci, V., Fortunato, D. and Giannoni, F.: On the existence of multiple geodesics in staticspace-times, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), 79–102.


8. Candela, A., Giannoni, F. and Masiello, A.: Multiple critical points for indefinite functionalsand applications, J. Differential Equations 155 (1999), 203–230.

9. Candela, A. and Salvatore, A.: Light rays joining two submanifolds in space-times, J. Geom.Phys. 22 (1997), 281–297.

10. Candela, A., Masiello, A. and Salvatore, A.: Existence and multiplicity of normal geodesics inLorentzian manifolds, J. Geom. Anal. 10 (2000), 591–619.

11. Canino, A.: On p-convex sets and geodesics, J. Differential Equations 75 (1988), 118–157.12. Caponio, E. and Masiello, A.: Trajectories for relativistic particles under the action of an

electromagnetic force in a stationary space-time, Nonlinear Anal. 50 (2002), 71–89.13. Fadell, E.: Lectures in cohomological index theories of G-spaces with applications to critical

point theory, Raccolta di seminari, Universitá della Calabria, 1985.14. Fadell, E. and Husseini, S.: Category of loop spaces of open subsets in Euclidean space,

Nonlinear Anal. 17 (1991), 1153–1161.15. Fournier, G. and Willem, M.: Relative category and the calculus of variations, In: H. Beresty-

cki, J. M. Coron and I. Ekeland (eds), Proc. ‘Variational Methods’, Birkhäuser, Basel, 1990,pp. 95–104.

16. Giannoni, F. and Masiello, A.: On the existence of geodesics on stationary Lorentz manifoldswith convex boundary, J. Funct. Anal. 101(2) (1991), 340–369.

17. Grove, K.: Condition (C) for the energy integral on certain path spaces and applications to thetheory of geodesics, J. Differential Geom. 8 (1973), 207–223.

18. Klingenberg, W.: Riemannian Geometry, De Gruyter, Berlin, 1982.19. Molina, J.: Alcune applicazioni della Teoria di Morse a varietá di Lorentz, PhD thesis, Univ.

Pisa, 1996.20. Nash, J.: The embedding problem for Riemannian manifolds, Ann. of Math. 63 (1956), 20–63.21. O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New

York, 1983.22. Rabinowitz, P. H.: Minimax Methods in Critical Point Theory with Applications to Differential

Equations, CBMS Regional Conf. Ser. Math. 65, Amer. Math. Soc., Providence, 1984.23. Sachs, R. K. and Wu, H.: General Relativity for Mathematicians, Springer, New York, 1977.24. Schwartz, J. T.: Nonlinear Functional Analysis, Gordon and Breach, New York, 1969.25. Szulkin, A.: A relative category and applications to critical point theory for strongly indefinite

functionals, Nonlinear Anal. 15 (1990), 725–739.


145

Asymptotic Distribution of Eigenvalues fora Class of Second-Order Elliptic Operatorswith Irregular Coefficients in R

d

LECH ZIELINSKILMPA, Université du Littoral, B.P. 699, 62228 Calais Cedex, France.e-mail: [email protected] and IMJ, Mathématiques, case 7012,Université Paris 7, 2 Place Jussieu, 75251 Paris Cedex 05, France

(Received: 11 October 2001; in final form: 29 March 2002)

Abstract. Let A = A0 + v(x) where A0 is a second-order uniformly elliptic self-adjoint operatorin Rd and v is a real valued polynomially growing potential. Assuming that v and the coefficientsof A0 are Hölder continuous, we study the asymptotic behaviour of the counting function N (A, λ)

(λ → ∞) with the remainder estimates depending on the regularity hypotheses. Our strongestregularity hypotheses involve Lipschitz continuity and give the remainder estimate N (A, λ)O(λ−µ),where µ may take an arbitrary value strictly smaller than the best possible value known in the smoothcase. In particular, our results are obtained without any hypothesis on critical points of the potential.

Mathematics Subject Classification (2000): 35P20.

Key words: spectral asymptotics, Weyl formula, Schrödinger operator, elliptic operator, pseudodif-ferential operator.

1. Introduction

This paper is devoted to a study of a self-adjoint operator in L2(Rd),

A = A0 + V, (1.1)

where A0 = −� or more generally A0 is a second-order differential operatoruniformly elliptic on R

d and V is the operator of multiplication by a polyno-mially growing function v. The operator A is bounded from below, its spectrumis discrete and we are interested in the asymptotic behaviour of the associatedcounting function N (A, λ), defined as the number of eigenvalues (counted withtheir multiplicities) smaller than λ.

Numerous works (cf., e.g., [1, 6, 26, 28] and references therein) show thatvery weak hypotheses on the potential v are sufficient to establish the followingasymptotic formula:

N (A, λ) ∼ (2π)−d

∫a(x,ξ)<λ

dx dξ,

where λ → ∞ and a(x, ξ) = |ξ |2 + v(x) in the case A0 = −�.

146 LECH ZIELINSKI

In this paper, we consider estimates of the form∣∣∣∣N (A, λ) − (2π)−d

∫a(x,ξ)<λ

dx dξ

∣∣∣∣ � Cλ−µN (A, λ) (1.2)

with a certain µ > 0, which are considered, e.g., in [5, 6, 10, 11] (cf. also [18,22–24, 34] containing similar results for boundary value problems).

Concerning the smooth case, we remark that the most precise spectral asymptot-ics have been obtained by means of microlocal analysis (cf., e.g., [12–14, 16, 17])and usually one can find an optimal value of µ which cannot be improved in (1.2)(e.g., the explicit computation of eigenvalues for the harmonic oscillator shows that(1.2) holds if and only if µ � 1). Concerning the problem we want to study, the es-timates (1.2) with optimal values of µ have been proved in particular in [17, 25, 31]and we will present the corresponding results in details in the second part of thissection. We note also that estimates (1.2) with smaller values of µ allow us to treatmore general smooth operators in R

d by other methods in [2, 4, 15, 21, 27, 29].The aim of this paper is to establish estimates (1.2) using Hölder continuity

hypotheses in the spirit of [32, 34], i.e. the operator of multiplication by a Höldercontinuous function is replaced by a suitable pseudodifferential operator and thenew problem can be investigated by means of a microlocal analysis. Similarly asin [30] and [34], our assumptions allow us to establish (1.2) with µ by taking anarbitrary value strictly smaller than the optimal value in the smooth situation. Theprincipal difficulty comes from the fact that a covering by conical neighbourhoodsused in the elliptic case considered in [34] must be refined similarly as in the paperof A. Mohamed [25].

We note that minor changes (e.g., the support of the Fourier transform of f ±n,λ

from Section 3 should be included in the interval [−ca−1n h−1

n ; ca−1n h−1

n ]) allow todevelop the approach of [35] and to establish (1.2) with the optimal value of µunder slightly stronger regularity hypotheses. However, for simplicity, this paperwill not treat the case of the optimal value of µ.

1.1. GENERAL DEFINITION OF A

We denote 〈x〉 = (1+|x|2)1/2 and assume that v is a measurable function satisfying

〈x〉c � v(x) � C〈x〉C (1.3)

for certain constants C, c > 0.We denote Dj = −i∂/∂xj and formally write

A0 =∑

1�j,k�d

Dj(aj,k(x)Dk),

where aj,k = ak,j ∈ L∞(Rd) are such that

a0(x, ξ) :=∑

1�j,k�d

aj,k(x)ξj ξk � c|ξ |2 (1.4)

holds for a certain c > 0.

ASYMPTOTIC DISTRIBUTION OF EIGENVALUES 147

These hypotheses allow us to define the self-adjoint operator A = A0 + V viathe positive quadratic form given by the formula

A[ϕ,ψ] =∫

Rd

( ∑1�j,k�d

aj,k(x)∂ϕ(x)

∂xj

∂ψ(x)

∂xk+ v(x)ϕ(x)ψ(x)

)dx

for ϕ,ψ ∈ C∞0 (Rd) and clearly the resolvent of A is compact.

1.2. REGULARITY HYPOTHESES

We fix 0 < r � 1, 0 < ρ � 1 − (r/2) and assume

|x − y| � c v(x)ρ �⇒ C−1v(x) � v(y) � Cv(x), (1.5)

|x − y| � 1 �⇒ |v(x) − v(y)| � C|x − y|r v(x)1−ρ, (1.6)

|x − y| � 1 �⇒ |aj,k(x) − aj,k(y)| � C|x − y|r v(x)−ρ (1.7)

for some C, c > 0.

THEOREM 1.1. We assume the hypotheses (1.3)–(1.7) and denote

a(x, ξ) := a0(x, ξ) + v(x), (1.8)

h(x, ξ) := v(x)−ρ′ 〈ξ 〉−r ′, (1.9)

where ρ ′, r ′ are arbitrary positive numbers satisfying ρ ′ < ρ and r ′ < r.If the constant C > 0 is large enough, then we have the estimate∣∣∣∣N (A, λ) − (2π)−d

∫a<λ

dx dξ

∣∣∣∣ � C

∫a(1−Ch)<λ<a(1+Ch)

dx dξ. (1.10)

1.3. COMMENTS

We describe some consequences of Theorem 1.1. First of all, we note that perform-ing the integration with respect to ξ , we find∫

a(x,ξ)<λ

dx dξ =∫v(x)<λ

(λ − v(x))d/2 ω(x) dx, (1.11)

where

ω(x) =∫a0(x,ξ)<1

dξ.

Therefore, introducing the function

V(λ) :=∫v(x)<λ

dx, (1.12)

148 LECH ZIELINSKI

we find that∫a<λ

dx dξ � C0λd/2V(λ) holds for a certain constant C0 > 0 and it is

easy to see that the additional assumption (cf. [28])

V(2λ) � CV(λ) (1.13)

implies

C−10 λd/2V(λ) �

∫a<λ

dx dξ � C0λd/2V(λ). (1.14)

Moreover, we have the following proposition:

PROPOSITION 1.2. Assume that (1.13) holds and the dimension d � 3. Then forevery C > 0, one can find C > 0 such that∫

a(1−Ch)<λ<a(1+Ch)

dx dξ

� Cλ(d−r ′ )/2

(λ−ρ′

V(λ) +∫v(x)<λ

v(x)−ρ′dx

). (1.15)

Remark 1.3. Similar but more complicated estimates are described in Section 9in the case of the dimension d = 1 and 2.

Let us discuss the important special case when the potential satisfies

〈x〉m � v(x) � C〈x〉m, (1.16)

with certain constants C,m > 0.It is easy to see that (1.16) ensures the estimates

C−10 λd/m � V(λ) � C0λ

d/m,∫v(x)<λ

v(x)−ρ′dx � C0λ

−ρ′+d/m,

with a certain C0 > 0, hence (1.10) with (1.15) imply∣∣∣∣N (A, λ) − (2π)−d

∫a<λ

dx dξ

∣∣∣∣ � Cλ−µ+d(1/2+1/m) (1.17)

with µ = (r ′/2) + ρ ′ and due to (1.14)

C−10 λd(1/2+1/m) �

∫a<λ

dx dξ � C0λd(1/2+1/m). (1.18)

Of course, (1.18) still holds if we put N (A, λ) instead of∫a<λ

dx dξ , hence theestimate (1.17) can be also written in the form (1.2).

The smooth situation with v satisfying (1.16) was considered by H. Tamura[30, 31], who proved the estimate (1.17) with the optimal value µ = 1/2 + 1/massuming that

|∂αaj,k(x)| � Cα〈x〉−|α|, |∂αv(x)| � Cα〈x〉m−|α|, (1.19)

x · ∇v(x) � c|x|m, for |x| > c−1 (1.20)


hold with certain constants Cα, c > 0. The same estimate follows from the paperof A. Mohamed [25], who considered a general microhyperbolic condition

|∇v(x)| � c|x|m−1, for |x| > c−1 (1.20′)

instead of (1.20). Finally, the theory developed in the book of V. Ivrii [17] givesthe possibility of skipping the microhyperbolic condition (1.20′) and can be alsoapplied to study problems with irregular coefficients as described in [18].

Since the assumptions (1.16), (1.19), imply (1.5), (1.6), (1.7) with r = 1,ρ = 1/m, we find that Theorem 1.1 gives the estimates (1.17) for every µ strictlysmaller than the optimal value 1/2 + 1/m.

The plan of the paper is detailed in Section 2 and we end this introduction bynoting the possibility of considering irregular potentials. More precisely, let A beas in Theorem 1.1 and

A = A + V = A0 + V + V , (1.21)

where V is the operator of multiplication by the real-valued measurable function v

satisfying

|v(x)| � Cv(x)1−κ (1.22)

for certain 0 < κ � 1 and C > 0. Since (1.22) implies v1−κ � A1−κ in the sense ofquadratic forms (cf. [7]), we have A−CA1−κ � A � A+CA1−κ and the min-maxprinciple (cf. [26]) implies

N (A + CA1−κ, λ) � N (A, λ) � N (A − CA1−κ , λ). (1.23)

Then it is easy to find λ0 large enough to ensure

N (A + CA1−κ, λ) � N (A, λ − 2Cλ1−κ ),

N (A − CA1−κ, λ) � N (A, λ + 2Cλ1−κ ),

for λ � λ0 and the asymptotic behaviour of N (A, λ ± 2Cλ1−κ) can be describedby using (1.10) with λ ± 2Cλ1−κ instead of λ.

It is possible to consider other conditions on v ensuring a decomposition v =v1 + O(v1−κ) with v1 satisfying the hypotheses analogical to (1.5), (1.6) (cf.,e.g., [34]) and we refer to [3] concerning the question of estimating the error term∫λ(1−2Cλ−κ)<a<λ(1+2Cλ−κ)

dx dξ .

2. Plan of the Proof

Our proof of Theorem 1.1 is based on a reduction to a smooth problem. Moreprecisely, we consider two pseudodifferential operators P+ and P−, which are self-adjoint operators in L2(Rd) with discrete spectrum satisfying

P− � A � P+ (2.1)

150 LECH ZIELINSKI

in the sense of quadratic forms. Then the min-max principle (cf. [26]) gives theinequalities between the corresponding counting functions

N (P+, λ) � N (A, λ) � N (P−, λ) (2.2)

and it suffices to investigate the asymptotic behavior of N (P±, λ).Below in this section, we give the details concerning P± and a Weyl formula

for the class of operators of this type is formulated in Theorem 2.1. The proof ofTheorem 2.1 is based on a microlocal analysis and its first step is described inSection 3. In this step we replace the Weyl formula of Theorem 2.1 by suitablemicrolocal estimates formulated in Theorems 3.1 and 3.2.

To obtain microlocal estimates of Theorems 3.1 and 3.2, we need to constructa suitable approximation of e−itP± . This construction is presented in Section 4 andit can be seen as a version of constructing a parabolic parametrix if t is imaginarynegative (cf. [19]). In Section 5, we begin the investigation of the error that ap-pears when the approximation replaces e−itP± . In particular, simple commutationsof pseudodifferential operators with the unitary group e−itP± allow us to observethe fact that modulo negligible errors in our setting the microsupport is conservedby the group. In Section 6, we consider the commutator of e−itP± with xj (theoperator of multiplication by the j th coordinate) and in Section 7 we end the proofof Theorem 2.1, using the results of Section 6 in a reasoning similar to the integra-tion by parts described in Lemma 4.4. In Section 8, we present a construction ofoperators P±, completing the proof of Theorem 1.1 and Proposition 1.2 is provedin Section 9.

2.1. DEFINITION OF A CLASS OF WEIGHT FUNCTIONS

Our approach is based on a pseudodifferential calculus with symbols estimatesinvolving some special weight functions like |ξ |2 + v(x) for instance.

For x, ξ ∈ Rd and c > 0 we introduce the sets

B−x (c) = {y ∈ R

d : |x − y| < cv(x)ρ}, (2.3)

B+ξ (c) = {η ∈ R

d : |ξ − η| < c〈ξ 〉}, (2.3′)

and the metric gx,ξ : Rd × R

d → ]0; ∞[ given by the formula

gx,ξ (y, η) = v(x)−2ρ |y|2 + 〈ξ 〉−2|η|2. (2.4)

Then Mc(g) will denote the set of continuous functions m: Rd × R

d → ]0; ∞[satisfying

C−1(1 + |x| + |ξ |)−C � m(x, ξ) � C(1 + |x| + |ξ |)C, (2.5)

(y, η) ∈ B−x (c) × B+

ξ (c) �⇒ C−1 � m(x, ξ)

m(y, η)� C (2.6)

for a certain constant C > 0.We will prove the following theorem:


THEOREM 2.1. Let us fix c, c, σ > 0 and define h by (1.9). Let p ∈ Mc(g) besuch that

p(x, ξ) � c(1 + |x| + |ξ |)c, (2.7)

|∂βx ∂αξ p(x, ξ)| � Cα,β p(x, ξ)h(x, ξ)1+σ |β|〈ξ 〉|β|−|α| (2.8)

for every α ∈ Nd , β ∈ N

d \ {0}. We set

p(x, ξ) = 〈ξ 〉|∇ξp(x, ξ)| + h(x, ξ)p(x, ξ) (2.9)

and assume that p ∈ Mc(g). We assume, moreover, that

(y, η) ∈ B−x (c) × B+

ξ (c) �⇒ 4|∇ηp(y, η) − ∇ξp(x, ξ)| � p(x, ξ)〈ξ 〉−1, (2.10)

the estimate

|∂αξ p(x, ξ)| � Cα p(x, ξ)〈ξ 〉−|α| (2.10′)

holds for every α ∈ Nd \ {0} and p � Cp for some C > 0.

Let pW (x,D) denote the Weyl quantization of p, i.e. the operator acting onϕ ∈ C∞

0 (Rd) according to the formula

(pW (x,D)ϕ

)(x) = (2π)−d

∫ei(x−y)ξ p

(x + y

2, ξ

)ϕ(y) dy dξ.

Then its closure in L2(Rd) defines the self-adjoint operator P with discrete spec-trum and its counting function N (P, λ) satisfies∣∣∣∣N (P, λ) − (2π)−d

∫p<λ

dx dξ

∣∣∣∣ � C

∫p(1−Ch)<λ<p(1+Ch)

dx dξ (2.11)

for a certain constant C > 0.

Then Theorem 1.1 follows from Theorem 2.1 and

THEOREM 2.2. Assume that the hypotheses of Theorem 1.1 hold. Then it is pos-sible to find p+, p− ∈ C∞(Rd × R

d) such that p = p± satisfy the hypotheses ofTheorem 2.1 with c > 0 and σ > 0 small enough. Moreover, (2.1) holds with P±being the closure of pW± (x,D) in L2(Rd) and

|∂αξ (p± − a)(x, ξ)| � Cαa(x, ξ)h(x, ξ)1+σ 〈ξ 〉−|α| (2.12)

holds for every α ∈ Nd .

Indeed, the condition (2.12) implies

a(1 − C0h) � p± � a(1 + C0h),

152 LECH ZIELINSKI

hence using

a < λ < p± �⇒ a < λ < a(1 + C0h)

and

p± < λ < a �⇒ a(1 − C0h) < λ < a,

we find∣∣∣∣∫p±<λ

dx dξ −∫a<λ

dx dξ

∣∣∣∣ �∫a(1−C0h)<λ<a(1+C0h)

dx dξ. (2.13)

Moreover, for every C > 0, we can find C > 0 such that∫p±(1−Ch)<λ<p±(1+Ch)

dx dξ �∫a(1−Ch)<λ<a(1+Ch)

dx dξ, (2.14)

hence it is clear that (2.2) and (2.11) with p = p± imply (1.10).

2.2. REMARKS CONCERNING WEIGHT FUNCTIONS

It is easy to check

m, m ∈ Mc(g) �⇒ m + m, mm, m/m ∈ Mc(g). (2.15)

Further on, we assume c < 1, which ensures the fact that m(x, ξ) = 〈ξ 〉s belongsto Mc(g) for every s ∈ R. Moreover, due to (1.5), we can assume that c > 0 issmall enough to ensure v ∈ Mc(g). In particular, h and 1 + |ξ |2 + v(x) belong toMc(g) and since there exist constants C, c > 0 such that

c(1 + |ξ |2 + v(x)) � a(x, ξ) � C(1 + |ξ |2 + v(x)), (2.16)

it is clear that a ∈ Mc(g).The next crucial point of our approach is the fact that

c|ξ | � |∇ξ a(x, ξ)| � C|ξ | (2.17)

holds for some C, c > 0. Indeed, to obtain the left inequality (2.17) it suffices touse the homogeneity of a0 with respect to ξ , writing

c|ξ |2 � 2a0(x, ξ) = ξ · ∇ξ a0(x, ξ) � |ξ ||∇ξa0(x, ξ)|.The properties of a imply the following useful fact: in order to find suitable p±defining P± in (2.1), it suffices to guarantee the estimates (2.8) and (2.12). Indeed,other hypotheses on p made in Theorem 2.1 follow thanks to


LEMMA 2.3. Assume that the hypotheses of Theorem 1.1 hold. Assume that psatisfies the estimates (2.8) for every α ∈ N

d , β ∈ Nd \ {0} and

|∂αξ (p − a)(x, ξ)| � Cαa(x, ξ)h(x, ξ)1+σ 〈ξ 〉−|α| (2.18)

for every α ∈ Nd . Then p ∈ Mc(g) and the formula (2.9) defines p ∈ Mc(g) such

that p � Cp, (2.10) and (2.10′) hold for every α ∈ Nd \ {0}.

Proof. Let p satisfy (2.18) and (2.8) for α ∈ Nd , β ∈ N

d \ {0}.Step 1. We check that p ∈ Mc(g). Due to |p − a| � C0ah

1+σ there is C > 0such that

|x| + |ξ | � C �⇒ C0h(x, ξ) � 1/2 �⇒ 1/2 � (p/a)(x, ξ) � 2.

By continuity of the quotient there is C > 0 such that C−1 < a/p < C anda ∈ Mc(g) ⇒ p ∈ Mc(g).

Step 2. We define p by (2.9) and check that p ∈ Mc(g), p � Cp. Since C−1 <

a/p < C and 〈ξ 〉|∇ξ (p − a)| � C1hp, introducing

a(x, ξ) := 〈ξ 〉|∇ξ a(x, ξ)| + (ha)(x, ξ), (2.19)

we obtain

a � 〈ξ 〉|∇ξp| + (C + C1)hp � C2p, (2.20)

p � 〈ξ 〉|∇ξ a| + (C + C1)hp � C2a. (2.20′)

However, due to (2.17) and ha � 1, we can find C > 0 such that

C−1(1 + |ξ |2 + ha) � a � C(1 + |ξ |2 + ha),

hence applying (2.20), (2.20′) we can find C > 0 such that

C−1(1 + |ξ |2 + ha) � p � C(1 + |ξ |2 + ha). (2.21)

Since 1 + |ξ |2 + ha ∈ Mc(g), (2.21) ensures p ∈ Mc(g). Finally a � Ca ⇒p � C ′p.

Step 3. We check that (2.10′) holds for every α ∈ Nd \ {0}. Since (2.21) implies

〈ξ 〉2 � Cp(x, ξ), (2.21′)

it is clear that for α ∈ Nd \ {0} we have

|∂αξ a(x, ξ)| � C〈ξ 〉2−|α| � CCp(x, ξ)〈ξ 〉−|α| (2.22)

and using (2.18) with hp � p, we obtain

|∂αξ (p − a)| � Cα〈ξ 〉−|α|h1+σp � Cα〈ξ 〉−|α|hσ p. (2.23)

Therefore combining (2.22) and (2.23) we obtain (2.10′).

154 LECH ZIELINSKI

Step 4. We show that (2.10) holds if c > 0 is small enough. Further on weassume y ∈ Bx(c). Then (1.7) implies

|y − x| � 1 �⇒ |aj,k(y) − aj,k(x)| � Ccrv(x)(r−1)ρ � Ccr . (2.24)

Next we check that

y ∈ Bx(c) �⇒ |aj,k(y) − aj,k(x)| � C(1 + |y − x|)v(x)−ρ . (2.25)

Indeed, it suffices to choose n ∈ N such that |y − x| � n � 1 + |y − x| and notethat (1.5)–(1.7) allow us to estimate |aj,k(y) − aj,k(x)| by

n∑k=1

|aj,k(y + k(x − y)/n) − aj,k(y + (k − 1)(x − y)/n)| � Cnv(x)−ρ.

Using (2.25), we can affirm that under the hypothesis y ∈ Bx(c) we have

|y − x| � 1 �⇒ |aj,k(y) − aj,k(x)| � 2C|y − x|v(x)−ρ � 2Cc � 2Ccr (2.26)

and, combining (2.24), (2.26), (2.21′), we obtain

|∇ξ a(y, ξ) − ∇ξ a(x, ξ)| � Ccr |ξ | � CCcr p(x, ξ)〈ξ 〉−1. (2.27)

If, moreover, η ∈ B+ξ (c), then

|∇ηa(x, η) − ∇ξ a(x, ξ)| � C|η − ξ | � Cc〈ξ 〉 � CCcr p(x, ξ)〈ξ 〉−1 (2.28)

and, choosing cr � 1/(16CC), we obtain

(y, η) ∈ B−x (c) × B+

ξ (c) �⇒ |∇ηa(y, η) − ∇ξa(x, ξ)| � 18 p(x, ξ)〈ξ 〉−1. (2.29)

If C0 > 0 is large enough to ensure Cαh(x, ξ)σ � 1

16 for |x| + |ξ | � C0 and|α| = 1 in the right-hand side of (2.23), then (2.29) ensures (2.10) for |x| +|ξ | � C0. It remains to note that the region {|x| + |ξ | � C0} is compact, hence(2.10) holds for all (x, ξ) ∈ R

2d if c > 0 small enough. ✷

3. Microlocal Partition of Unity

To begin we precise some details of our notations.We write R

∗ = R\{0}, N∗ = N\{0} and, for Z ⊂ R

k (k ∈ N∗), we denote by 1Z

the characteristic function of Z defined on Rk. We denote byB(L2(Rd)) the algebra

of bounded operators on L2(Rd) and ||·|| is the norm of B(L2(Rd)). If Q is of traceclass on L2(Rd), then its trace norm ||Q||tr = tr(QQ∗)1/2, where Q∗ denotes theadjoint of Q. Moreover b(x,D) denotes the standard pseudodifferential operatoracting according to the formula

(b(x,D)ϕ)(x) = (2π)−d

∫eixξ b(x, ξ)ϕ(ξ) dξ,


where ϕ is the Fourier transform of ϕ ∈ C∞0 (Rd).

Due to (1.5) the metrics g−x (y) = v(x)−2ρ |y|2 and g+

ξ (η) = 〈ξ 〉−2|η|2 are slowlyvarying in the sense of the Definition 18.4.1 in [16]. Thus Lemma 18.4.4 in [16]ensures that for any c > 0 small enough one can choose a sequence {x(k)}k∈N∗(respectively {ξ (k)}k∈N∗) of points in R

d giving a covering of Rd by the family

{B−x(k)(c)}k∈N∗ (respectively {B+

ξ (k)(c)}k∈N∗) satisfying

∞∑k=1

1B−x(k)

(c)(x) � Nc and∞∑k=1

1B+ξ (k)

(c)(ξ ) � Nc (3.1)

for a certain Nc ∈ N. Moreover, we can find real valued functions θ−k ∈

C∞0 (B−

x(k)(c/2)), θ+k ∈ C∞

0 (B+ξ (k)

(c/2)) satisfying

∞∑k=1

(θ−k )

2 =∞∑k=1

(θ+k )

2 = 1,

|∂αθ−k (x)| � Cαv(x)

−ρ|α|, |∂αθ+k (ξ)| � Cα〈ξ 〉−|α|. (3.2)

Let 5−k denote the operator of multiplication by θ−

k and 5+k = θ+

k (D). Then

I =∞∑

k−=1

∞∑k+=1

5+k+(5

−k−)

25+k+ =

∞∑n=1

L∗nLn, (3.2′)

where n → (k−n , k

+n ) is a fixed bijection N

∗ → N∗ × N

∗ and Ln = 5−k−n5+

k+n

=ln(x,D) with

ln(x, ξ) = θ−k−n(x)θ+

k+n(ξ).

Let λ ∈ R. Since 1]−∞, λ[ denotes the characteristic function of the interval ]−∞, λ[,the corresponding spectral projector of P can be written as 1]−∞, λ[(P ) and dueto (3.2′) and the trace cyclicity,

N (P, λ) = tr 1]−∞, λ[(P ) =∞∑n=1

trLn 1]−∞, λ[(P )L∗n. (3.3)

Expression (3.3) allows us to replace Theorem 2.1 by

THEOREM 3.1. Let N0 ∈ N. Then one can find a constant C = C(N0) such thatfor every n ∈ N

∗ one has the estimate∣∣∣∣ trLn 1]−∞; λ[(P )L∗n − (2π)−d

∫p<λ

ln(x, ξ)2 dx dξ

∣∣∣∣� C

∫λ−pnhn<p<λ+pnhn

1Bn(c)(x, ξ) dx dξ + ChN0n , (3.4)

156 LECH ZIELINSKI

where we have denoted

pn := p(x(k−n ), ξ (k

+n )), hn := h(x(k−

n ), ξ (k+n )),

Bn(c) := B−x(k−

n )(c) × B+

ξ (k+n )(c).

Let us check that Theorem 3.1 implies Theorem 2.1. Due to (3.3) and∑∞

n=1 l2n =

1, it suffices to show that for N0 ∈ N large enough, the quantity∞∑n=1


1Bn(c)(x, ξ) dx dξ +∞∑n=1

hN0n (3.5)

can be estimated by the right-hand side of (2.11).If m ∈ Mc(g) and mn = m(x(k−

n ), ξ (k+n )), then

(x, ξ) ∈ Bn(c) �⇒ C−1mn � m(x, ξ) � Cmn. (3.6)

In particular for (x, ξ) ∈ Bn(c) we have the implication

λ − pnhn < p(x, ξ) < λ + pnhn �⇒ p(1 − Ch)(x, ξ) < λ < p(1 + Ch)(x, ξ)

and due to (3.1) we find the desired estimate of the first sum of (3.5).To estimate the second sum of (3.5), we note that vol(Bn(c)) � c0 > 0 and

∞∑n=1

hN0n �

∞∑n=1

c−10 hN0

n vol(Bn(c)) � C

∫h(x, ξ)N0 dx dξ < ∞

if N0 is large enough.

3.1. APPROXIMATION OF THE CHARACTERISTIC FUNCTION 1]−∞,λ[

Let f+ ∈ C∞0 (]0; 1[) and f− ∈ C∞

0 (]−1; 0[) be such that∫∞

−∞ f± = 1 andf± � 0. For λ ∈ R and n ∈ N

∗ we define

f ±n,λ(s) :=

∫ ∞

s

f±(τ − λ

hnpn

)dτ

hnpn

. (3.7)

Then we have

1]−∞, λ−hnpn[ � f −n,λ � 1]−∞, λ[ � f +

n,λ � 1]−∞, λ+hnpn[. (3.8)

Moreover, for k � 1, the kth derivative of f ±n,λ satisfies

supp(f ±n,λ)

(k) ⊂ [λ − hnpn; λ + hnpn],|(f ±

n,λ)(k)| � Ckh

−kn p−k

n . (3.9(k))

Let f ±n,λ denote the inverse Fourier transform of f ±

n,λ. Then

t f ±n,λ(t) = ieitλf±(hnpnt), (3.10)

where f± ∈ S(R) is the inverse Fourier transform of f ±.


Due to (3.10), sing supp f ±n,λ ⊂ {0} and for every N ∈ N we have

|t f ±n,λ(t)| � CN〈hnpnt〉−N. (3.11)

The inequalities (3.8) imply

trLn f−n,λ(P )L

∗n � trLn 1]−∞, λ](P )L∗

n � trLn f+n,λ(P )L

∗n (3.12)

and it suffices to prove that (3.4) holds with trLn 1]−∞, λ](P )L∗n replaced by

trLn f±n,λ(P )L

∗n =

∫ ∞

−∞dt f ±

n,λ(t) trLn e−itP L∗n, (3.13)

where the integral in a neighbourhood of 0 can be considered as the distributionf ±n,λ acting on the smooth function t → trLn e−itPL∗

n.To recover the behaviour of trLn e−itP L∗

n we will construct a sequence of op-erators {QN,n(t)}N∈N

being a suitable approximation of Lne−itP . More precisely:instead of Theorem 3.1 it suffices to prove Theorem 3.2.

THEOREM 3.2. There exists a sequence of operators {QN,n(t)}N∈Nsuch that∣∣∣∣

∫ ∞

−∞dt f ±

n,λ(t)(trLn e−itP L∗

n − trQN,n(t)L∗n

)∣∣∣∣ � ChN0n , (3.14)∣∣∣∣

∫ ∞

−∞dt f ±

n,λ(t) trQN,n(t)L∗n − (2π)−d

∫p<λ

ln(x, ξ)2 dx dξ

∣∣∣∣� C


1Bn(c)(x, ξ) dx dξ (3.15)

hold if N = N(N0) ∈ N and C = C(N,N0) > 0 are large enough.

Further on we denote [0; t] = {st ∈ R : 0 � s � 1} and

V = {(n, t, τ ) : n ∈ N∗, t ∈ R

∗, τ ∈ [0; t]}. (3.16)

Introducing

QN,n(t) = d

dtQN,n(t) + iQN,n(t)P

and assuming QN,n(0) = Ln, we can formally write

Lne−itP − QN,n(t) =∫ t

0dτ

d

dτ

(QN,n(t − τ)e−iτP

)= −

∫ t

0dτQN,n(t − τ)e−iτP , (3.17)

hence in order to obtain the estimates (3.14), it suffices to prove the existence of aconstant C = C(N,N0) (independent of (n, t, τ )) such that∣∣tr QN,n(t − τ)e−iτP L∗

n

∣∣ � ChN0n 〈hnpnt〉C (3.18)

holds for (n, t, τ ) ∈ V.

158 LECH ZIELINSKI

The construction of QN,n(t) presented in the next section uses pseudodiffer-ential operators and the end of this section is devoted to a description of suitableclasses of symbols.

3.2. CLASSES OF SYMBOLS

We consider Rd × R

d equipped with the metric

gx,ξ (y, η) = h(x, ξ)2σ 〈ξ 〉2|y|2 + 〈ξ 〉−2|η|2.If m ∈ Mc(g) then S(m, g) denotes the class of functions b ∈ C∞(Rd × R

d)

satisfying

|∂βx ∂αξ b(x, ξ)| � Cα,β m(x, ξ)h(x, ξ)σ |β|〈ξ 〉|β|−|α| (3.19)

for every α, β ∈ Nd .

We recall some basic properties of S(m, g) described in Ch. 18 of [16]:

PROPOSITION 3.3. Let b ∈ S(m, g) and b ∈ S(m, g). Then

bb ∈ S(mm, g), |b| � m �⇒ 1/b ∈ S(1/m, g) (3.20)

and there exist b0 ∈ S(m, g), b � b ∈ S(mm, g) such that

bW (x,D) = b0(x,D), bW (x,D)bW (x,D) = (b � b)W (x,D). (3.21)

Moreover,

∂xj ∂ξj b ∈ S(m′, g) (j = 1, . . . , d) �⇒ b − b0 ∈ S(m′, g), (3.22)

∂ξj b ∈ S(m′, g), ∂xj b ∈ S(m′′, g), ∂ξj b ∈ S(m′, g),

∂xj b ∈ S(m′′, g) (j = 1, . . . , d) �⇒ b � b − bb ∈ S(m′m′′ + m′′m′, g). (3.23)

We remark that the inequalities (2.8), (2.18) still hold if their left-hand sidescontain p + r with r ∈ S(h1+σp, g) instead of p + r. In particular we have

COROLLARY 3.4. If p satisfies (2.8), then using (3.22) with b = p, and m =h1+σp, we find p0 such that

pW (x,D) = p0(x,D) and p − p0 ∈ S(h1+σp, g). (3.24)

Moreover, the estimates (2.8), (2.18) hold with p0 instead of p in the left-hand sideof each inequality.

In Section 4 we consider sequences of symbols {bn}n∈N∗ and later on the familiesof symbols {bν}ν∈V . We adopt the convention that the notation bn ∈ S(m, g) orbν ∈ S(m, g) always means that the estimates (3.19) hold with bn or bν instead ofb with constants Cα,β independent of n ∈ N

∗ or ν ∈ V.


4. Construction of the Approximation

In this section N ∈ N is fixed and we describe the construction of

QN,n(t) = (e−itpqN,n(t)

)(x,D) (4.25)

such that (3.15) holds. The proof of (3.14) will be given in Sections 5–7.For N = 0, . . . , N we are going to consider

qN,n(t)(x, ξ) = qN,n(t, x, ξ) =∑

0�k�N

tkq 0k,n(x, ξ), (4.2(N))

where qN,n(0) = q 00,n = ln and for k � 1 we have

q 0k,n ∈ S(hp(hσ/2p)k−1, g), supp q 0

k,n ⊂ Bn(c/2). (4.3(k))

We construct qN,n(t) by induction with respect toN and its final step correspondingto N = N gives qN,n(t) to be used in (4.1).

For a smooth function (t, x, ξ) → b(t, x, ξ) ∈ C we denote

PN b(t) =∑

|α|�N

(−i)|α|∂aξ(b(t)∂αx p0

)/α!,

where p0 is given by (3.24).

PROPOSITION 4.1. Let N = 0, . . . , N . Then we can find qN,n(t) satisfying(4.2(N)) with qN,n(0) = ln, q 0

k,n satisfying (4.3(k)) for k = 1, . . . , N and

(∂t + iPN )(e−itpqN,n(t)

) = e−itpq 0N,n(t) (4.4(N))

holds with

q 0N,n(t)(x, ξ) =

∑N�k�N+N

tk q 0N,k,n(x, ξ), (4.5(N))

q 0N,k,n ∈ S(hp(hσ/2p)k, g) for k = N, . . . , N + N (4.6(N))

and supp q 0N,k,n ⊂ Bn(c/2).

Proof. We introduce the notation

PN q(t) = eitp(∂t + iPN )(e−itpq(t)

).

If q(t, x, ξ) = q0(x, ξ) is independent of t , then

PN q(t) =∑

0�k�N

tk q 0k (4.7)

160 LECH ZIELINSKI

with

q 00 = i(p0 − p)q0 −

∑1�|α|�N

(−i)|α|+1∂αξ (q0 ∂αx p0)/α!,

q 0k =

∑|α0+···+αk |�N

αj !=0 if j !=0

cα0,...,αk∂α0ξ (q0 ∂

α0+···+αkx p0)∂

α1ξ p . . . ∂

αkξ p (k = 1, . . . , N).

Using the fact that the estimates (2.8) still hold with p0 instead of p, we find

q0 ∈ S(m, g) �⇒ ∂α0ξ (q0 ∂αx p0) ∈ S(mph1+σ |α|〈ξ 〉|α|−|α0|, g) (4.8)

if α != 0. Moreover, the estimates (2.10′) imply

∂α1ξ p . . . ∂

αkξ p ∈ S(pk〈ξ 〉−|α1|−···−|αk |, g) (4.9)

if α1, . . . , αk != 0. Combining (4.8), (4.9) with α = α0 + · · · + αk ⇒ |α| �max{k, 1} in the expression of q 0

k we find

q0 ∈ S(m, g) �⇒ q 0k ∈ S(mph1+σ max{k, 1}pk, g) (4.10)

(where in the case k = 0 we use, moreover, (3.24)).Therefore in the case N = 0 when we take q(t) = ln ∈ S(1, g) in (4.7), we

obtain (4.4(0)) with q 00,n of the form (4.5(0)) and (4.10) with m = 1 imply (4.6(0)),

i.e. Proposition 4.1 holds for N = 0.Further on, we assume that the statement of Proposition 4.1 holds for a given

N � N − 1 and we prove that it still holds for N + 1 instead of N .Using the induction hypotheses (4.4(N)), (4.5(N)) to express PNqN,n(t) we find

PN qN+1,n(t)

= PN (tN+1q 0

N+1,n) + PN qN,n(t)

= tN((N + 1)q 0

N+1,n + q 0N,N,n

)+tN+1PNq0N+1,n +

∑N+1�k�N+N

tk q 0N,k,n.

In order to obtain (4.5(N + 1)), it suffices to cancel the term with tN taking

q 0N+1,n = −q 0

N,N,n/(N + 1) (4.11)

and q 0N,N,n ∈ S(hp(hσ/2p)N , g) by the induction hypothesis (4.6(N)).

Let us introduce the following notation:

an(t) ∈∑k∈K

tkS(m(k), g) ⇐⇒ an(t) =∑k∈K

tka 0k,n with a 0

k,n ∈ S(m(k), g).

Then using ph � p and max{k, 1} � (k + 1)/2, we can write (4.7), (4.10) in thefollowing form:

q(t) = q0 ∈ S(m, g) �⇒ PNq(t) ∈∑

0�k�N

tkS(m(hσ/2p)k+1, g

). (4.12)


Since (4.11) gives (4.3(N + 1)), applying (4.12) with m = ph(hσ/2p)N , we find

tN+1PN q0N,n+1 ∈

∑0�k�N

tN+1+k S(ph(hσ/2p)N+k+1, g)

and (4.6(N + 1)) follows. ✷PROPOSITION 4.2. Let QN,n(t) be defined by (4.1) with qN,n given by (4.2(N))

where q 0n,0 = ln and q 0

k,n satisfy (4.3(k)) for k = 1, . . . , N . Then (3.15) holds.Proof. For k ∈ N, n ∈ N

∗, q ∈ C∞0 (Rd × R

d) and λ ∈ R we denote

Nk,n(q, λ) =∫ ∞

−∞dt f ±

n,λ(t)tkJt (q), (4.13)

where

Jt (q) = (2π)−d

∫e−itp(x,ξ)q(x, ξ) dx dξ. (4.14)

Since tkf ±n,λ(t) is the Fourier inverse of ik(f ±

n,λ)(k), changing the order of inte-

grals (4.13) and (4.14), we find

Nk,n(q, λ) = (2π)−d

∫ik(f ±

n,λ)(k)(p(x, ξ)

)q(x, ξ) dx dξ. (4.15)

Since (e−itpq)(x,D)L∗n has the integral kernel

(x, y) → (2π)−d

∫e−itp(x,ξ)q(x, ξ)ln(y, ξ) dξ,

we have trL∗n(e

−itpq)(x,D) = Jt (qln) and∫ ∞

−∞dt f ±

n,λ(t) trQN,n(t)L∗n =

∑0�k�N

Nk,n(q0k,nln, λ). (4.16)

For k = 0, we have q 00,n = ln, hence (4.15) and (3.8) give∣∣∣∣N0,n(q

00,nln, λ) − (2π)−d

∫p<λ

ln(x, ξ)2 dx dξ

∣∣∣∣=∣∣∣∣(2π)−d

∫ (f ±n,λ − 1]−∞,λ[

)(p(x, ξ)

)ln(x, ξ)

2 dx dξ

∣∣∣∣�∫λ−hnpn<p<λ+hnpn

1Bn(c)(x, ξ) dx dξ.

It remains to show that the estimate

|Nk,n(qn, λ)| � C

∫λ−hnpn<p<λ+hnpn

1Bn(c)(x, ξ) dx dξ (4.17(k))

holds with qn = q 0k,nln, k = 1, . . . , N .

162 LECH ZIELINSKI

The expression (4.15) allows us to write the obvious inequality

|Nk,n(qn, λ)| �∫ ∣∣(f ±

n,λ)(k)(p(x, ξ)

)∣∣|qn(x, ξ)| dx dξ. (4.18(k))

In the case k = 1, the above inequality leads to the following corollary:

COROLLARY 4.3. If qn ∈ S(ph, g) are such that supp qn ⊂ Bn(c), then |qn| �Chnpn and using (3.9(1)) in (4.18(1)), we obtain (4.17(1)). In particular, (4.17(1))holds with qn = q 0

1,nln.

In order to prove (4.17(k)) for qn = q 0k,nln, k � 2, let χ ∈ C∞

0 (]−2; 2[) besuch that χ = 1 on [−1; 1] and for s > 0 let

χs(x, ξ) = χ(m(x, ξ)/s2) with m = 〈ξ 〉2|∇ξp|2h−2p−2. (4.19)

We also consider χs := 1 − χs and remark that using 1 + m ∈ S(1 + m, g), it iseasy to check that χs, χs ∈ S(1, g). Moreover

(x, ξ) ∈ suppχs�⇒ p(x, ξ) = ((1 + m1/2)hp)(x, ξ) � (1 + 2s)(hp)(x, ξ). (4.20)

Therefore |q 0k,nlnχs | � Ckh

knp

kn and using (3.9(k)) in (4.18(k)) we find that

(4.17(k)) holds with qn = q 0k,nlnχs , k � 2.

Thus it remains to show that (4.17(k)) holds with qn = q 0k,nlnχs , k � 2.

To obtain this result, it suffices to show

LEMMA 4.4. If qn ∈ S(m, g), then we can find qn ∈ S(m/p, g) such thatsupp qn ⊂ supp qn and

tJt (qnχs) = Jt (qnχs/2). (4.21)

Indeed, iterating the assertion of Lemma 4.4, we can write

tkJt (q0k,nlnχs) = tk−1Jt(q1,k,nχs/2) = · · · = tk−kJt (qk,k,nχs/2k ) (4.22)

for some qk,k,n ∈ S(phpk−1−k, g) with supp qk,k,n ⊂ Bn(c/2). Thus using (4.22)with k = k − 1 we obtain

Nk,n(q0k,nlnχs, λ) = N1,n(qn, λ)

with qn ∈ S(hp, g), supp qn ⊂ Bn(c/2), and (4.17(k)) holds due to Corollary 4.3.

Proof of Lemma 4.4. Using 〈ξ 〉2|∇ξp|2 ∈ S(p2, g) and

s2p(x, ξ)2 � 2s2(hp)(x, ξ)2 + 2〈ξ 〉2|∇ξp(x, ξ)|2� 2(s2 + 1)〈ξ 〉2|∇ξp(x, ξ)|2


for (x, ξ) ∈ supp χs , we obtain χs〈ξ 〉−2|∇ξp|−2 ∈ S(p−2, g) due to (3.20) andcombining with 〈ξ 〉2∂ξj p ∈ S(〈ξ 〉p, g), we can define

χj,s := χs ∂ξj p|∇ξp|−2 ∈ S(〈ξ 〉/p, g). (4.23)

Writing qnχs = ∑dj=1 qnχj,s∂ξj p and integrating by parts, we find

tJt (qnχj,s∂ξj p) = (2π)−d

∫i∂ξj

(e−itp(x,ξ)

)(qnχj,s )(x, ξ) dx dξ

= −iJt (∂ξj (qnχj,s)),

which completes the proof due to ∂ξj (qnχj,s) ∈ S(m/p, g) and supp χj,s ∩supp χs/2 = ∅. ✷

5. Preliminary Remarks about the Approximation Error

In this section we begin a study of tr QN,n(t − τ)e−iτP L∗n with the purpose of

establishing (3.18), which implies (3.14).To abbreviate notations, we denote the elements of V by the letter ν, adopting

the following convention: if the index ν appears in a formula simultaneously witha letter n, t or τ , then ν = (n, t, τ ). Moreover, the notation sν = sν + O(mn,t )

means that |sn,t,τ − sn,t,τ | � Cmn,t holds with a constant C > 0 independent ofν = (n, t, τ ) ∈ V.

Assume that Qν = (e−itpqν)(x,D) with

qν ∈ S(m, g), supp qν ⊂ Bn(c) and Yν ∈ B(L2(Rd)) for ν ∈ V.

Then Lemma 9.1 allows us to estimate

|tk trQνe−iτP Yν | � |t|k‖Qν‖tr ‖Yν‖ � Cmnh−C0n 〈hnpnt〉k+C0‖Yν‖, (5.1)

where mn are as in (3.6).The family {Yν}ν∈V ⊂ B(L2(Rd)) will be called negligible, if for every N ∈ N,

we can find C(N) > 0 such that

‖Yν‖ = O(hNn 〈hnpnt〉C(N)

). (5.2)

If N ′, k ∈ N and {Yν}ν∈V is negligible, then (5.1) allows to find C(k,N ′) > 0 suchthat

tk trQνe−iτP Yν = O(hN

′n 〈hnpnt〉C(k,N ′)). (5.3)

PROPOSITION 5.1. Let c > c/2 and consider bν ∈ S(m, g) satisfying supp bν ∩Bn(c) = ∅. Then the family

Rν = bWν (x,D)e−iτP L∗

n (5.4)

is negligible.

164 LECH ZIELINSKI

Before giving the proof of this result we describe its consequences.We assume c > c > c′ > c/2 and note that the method of Ch. 18.4 in [16] (cf.

the beginning of Section 3) allows us to find θ−k ∈ C∞

0 (B−x(k)(c)) such that θ−

k = 1on B−

x(k)(c′) and

|∂αθ−n (x)| � Cαv(x)

−ρ|α|.

If 5−k denotes the operator of multiplication by θ−

k , then Proposition 5.1 ensuresthat (I − 5−

k−n)e−iτP L∗

n and P(I − 5−k−n)e−iτP L∗

n are negligible, hence for everyN ∈ N, we can find C(N) > 0 such that

tr QN,n(t − τ)e−iτP L∗n = tr QN,n(t − τ)5−

k−n

e−iτP L∗n + O

(hNn 〈hnpnt〉C(N)

). (5.5)

Further on, we prefer using the operators QN,n(t − τ)5−k−n

instead of QN,n(t − τ)

and for this purpose we introduce new notations.

5.1. SYMBOLS DEPENDING ON (x, ξ, y) ∈ Rd × Rd × Rd

If qν ∈ C∞0 (Rd × R

d × Rd), then Op(ei(τ−t )pqν) will denote the integral operator

with the kernel

(x, y) → Kν(q, x, y) = (2π)−d

∫ei(x−y)ξ+i(τ−t )p(x,ξ)qν(x, ξ, y) dξ. (5.6)

For c > 0 and n ∈ N∗, we will denote

Bn(c) := B−x(k−

n )(c) × B+

ξ (k+n )

× B−x(k−

n )(c)

and, for m ∈ Mc(g), we will write qν ∈ S(m) if and only if there is c < c such thatqν ∈ C∞

0 (Bn(c)) and

|∂βx,y∂αξ qν(x, ξ, y)| � Cα,β mnhσ |β|n 〈ξ 〉|β|−|α|

holds for every α ∈ Nd , β ∈ N

2d , where mn are as in (3.6).These notations will be used below to express the approximation error in a

particular form, similar to (4.13), (4.14), (4.16).

5.2. EXPRESSIONS OF tr QN,n(t − τ )5−k−n

e−iτpL∗n

We have

pW (x,D) = p0(x,D) = p0(x,D)∗

and

QN,n(t)p0(x,D)∗ = Op(e−itpqN,n,t

)with

qN,n,t (x, ξ, y) = qN,n(t)(x, ξ)p0(y, ξ).


Writing Taylor’s development of qN,n,t (x, ξ, ·) in x and applying standard integra-tions by parts in the integrals of the form (5.6) based on the equality

(x − y)αei(x−y)ξ = i|α|∂αξ(ei(x−y)ξ

),

we find

QN,n(t) = Op(e−itp(q 0

N,n+ q 1

N,n)(t)

),

where q 0N ,n

(t) are as in Proposition 4.1 and the remainder term of Taylor’s devel-

opment of order N gives

q 1N,n

(t, x, ξ, y) = eitp(x,ξ)(N + 1)∫ 1

0ds(1 − s)N qN,n(t, s, x, ξ, y), (5.7)

with

qN,n(t, s, x, ξ, y)

=∑

|α|=N+1

(−i)|α|(∂αξ (qN,n(t)e−itp

)(x, ξ)∂αx p0(x + s(y − x), ξ)

)/α!.

We can write

q 0N,n

(t − τ, x, ξ)θ−k−n(y) =

∑N�k�2N

(t − τ)kq 0N,k,n

(x, ξ)θ−k−n(y)

=∑

N�k�2N

tk q 0N,k,ν

(x, ξ, y),

where ν = (n, t, τ ) according to our convention and

q 0N,k,ν

(x, ξ, y) =(

1 − τ

t

)k

q 0N,k,n

(x, ξ)θ−k−n(y).

It is clear that q 0N ,k,ν

∈ S(phNσ/2pk) and using the form of q 0N ,n

(t) in (5.7), we finda similar expression

q 1N,n

(t − τ)(x, ξ, y)θ−k−n(y) =

∑0�k�2N

tkq 1N,k,ν

(x, ξ, y)

with q 1N,k,ν

∈ S(phNσ/2pk).

If Yν ∈ B(L2(Rd)) and qν ∈ S(m), then the quantity

Jν(q, Y ) := tr(Op(ei(τ−t )pqν)e

−iτP Yν)

(5.8)

is well defined due to Lemma 9.1 and, analogically to (5.1), we have

|tkJν(q, Y )| � Cmnh−C0n 〈hnpnt〉k+C0‖Yν‖, (5.9)

where mn are as in (3.6).Using the above notation, we can state the following conclusion:

166 LECH ZIELINSKI

COROLLARY 5.2. There exist qN,k,ν ∈ S(phNσ/2pk) such that

tr QN,n(t − τ)5−k−n

e−iτP L∗n =

∑0�k�2N

tkJν(qN,k, Y ) (5.10)

holds with Yν = L∗n.

We will complete the proof of Theorem 2.1 by showing the following proposi-tion:

PROPOSITION 5.3. Assume N0, k ∈ N∗, Yν = L∗

n and qν ∈ S(m). Then we canfind k0 ∈ N

∗ and C(N0) > 0 such that

tkJν(q, Y ) =∑

1�k�k0

Jν(qk, Yk) + O(hN0n 〈hnpnt〉C(N0)

)(5.11)

holds for certain families of symbols {qk,ν}ν∈V and operators {Yk,ν}ν∈V satisfying

qk,ν ∈ S(mp−k), ‖Yk,ν‖ = O(〈hnpnt〉C(N0)

)(5.12)

for k = 1, . . . , k0.

Indeed, using (5.10) and Proposition 5.3 with qν = qN,k,ν , m = phNσ/2pk, wecan write

tr QN,n(t − τ)5−k−n

e−iτP L∗n

=∑

1�k�k(N)

Jν(qN,k, YN,k) + O(hN0n 〈hnpnt〉C(N,N0)

),

with

qN,k,ν ∈ S(phNσ/2), ‖YN,k,ν‖ = O(〈hnpnt〉C(N,N0)

),

hence choosing N = N(N0) large enough, we can ensure

Jν(qN,k, YN,k) = O(hN0n 〈hnpnt〉C(N ,N0)

),

due to (5.9). This proves (3.18), completing the proof of Theorems 3.2 and 2.1.

Proof of Proposition 5.1. The method of Ch. 18.4 in [16] allows us to find thesymbols

l0n(x, ξ) = θ0 −k−n(x)θ0 +

k+n(ξ)

satisfying

l0n ∈ S(1, g), ∂xj l0n ∈ S(v(x)−ρ, g), supp l0n ⊂ Bn(c),

l0n = 1 on Bn(c′) with c > c′ > c/2.

(5.13)


We set L0n = l0n(x,D) and note that ‖bW

ν (x,D)L0n‖ = O(hNn ) for every N ∈ N

holds due to supp bν ∩ supp l0n = ∅ (cf. Theorems 18.5.4 and 18.6.3 in [16]).Therefore, it suffices to show that

‖(I − L0n)e

−iτP L∗n‖ = O

(hkσn 〈hnpnτ 〉k) (5.14(k))

holds for every k ∈ N if l0n satisfies (5.13). Obviously (5.14(k)) holds for k = 0and we will prove the general statement by induction with respect to k ∈ N.

Further on, we assume that σ > 0 is small enough to ensure ρ ′(1 + σ ) � ρ andr ′(1 + σ ) � 1, implying v(x)−ρ � 〈ξ 〉h(x, ξ)1+σ . Then

∂xj p ∈ S(〈ξ 〉ph1+σ , g), ∂ξj l0n ∈ S(〈ξ 〉−1, g),

∂ξj p ∈ S(〈ξ 〉−1p, g), ∂xj l0n ∈ S(v(x)−ρ, g) ⊂ S(〈ξ 〉h1+σ , g)

and (3.23) ensures

[I − L0n, P ] = −[L0

n, P ] = lWn (x,D) with ln ∈ S(ph1+σ , g). (5.15)

Moreover, we can find

l1n ∈ S(1, g), ∂xj l1n ∈ S(v(x)−ρ, g), supp l1n ⊂ Bn(c

′),l1n = 1 on Bn(c

′′) with c′ > c′′ > c/2(5.16)

and setting L1n = l1n(x,D), we have supp ln ∩ supp l1n = ∅, hence

‖lWn (x,D)L1n‖ = O(hNn ) for every N ∈ N. (5.17)

Since ‖(I − L0n)L

∗n‖ = O(hNn ) for every N ∈ N and

(I − L0n)e

−iτP L∗n = e−iτP (I − L0

n)L∗n + [

I − L0n, e−iτP

]L∗n, (5.18)

it remains to estimate the norm of

[I − L0

n, e−iτP]L∗n =

∫ 1

0dsτei(s−1)τP [I − L0

n, P ]e−isτPL∗n (5.19)

and due to (5.17) it suffices to estimate

|τ |‖lWn (x,D)(I − L1n)e

−isτPL∗n‖. (5.20)

However, we have ‖lWn (x,D)‖ � Cpnh1+σn (cf. Theorem 18.6.3 in [16]) and using

the induction hypothesis, we can assume that (5.14(k)) holds with L1n instead of L0

n

and c′′ instead of c′. Thus, the quantity (5.20) can be estimated by

C|τ |pnh1+σn hkσn 〈hnpnτ 〉k � Ch(k+1)σ

n 〈hnpnτ 〉k+1,

completing the proof of (5.14(k + 1)). ✷

168 LECH ZIELINSKI

6. Auxiliary Commutator Formulas

6.1. NOTATIONS

We will write bν ∈ Sn(m, g) if and only if bν ∈ S(m, g) and there exists c0 < c,l0n ∈ S(1, g) satisfying supp l0n ⊂ Bn(c0) and (1 − l0n)bν ∈ S(hN, g) for every N ∈N. Then Theorem 18.5.4 in [16] ensures bb, b � b ∈ Sn(mm, g) if bν ∈ Sn(m, g),bν ∈ S(m, g) and (3.23) still holds with b, b, S(m′m′′ + m′′m′, g) replaced bybν, bν , Sn(m

′m′′ + m′′m′, g). Moreover, bν ∈ Sn(m, g) implies

|∂βx ∂αξ bν(x, ξ)| � Cα,β mnhσ |β|n 〈ξ 〉|β|−|α|, (6.1)

where mn are as in (3.6) and Theorem 8.6.3 [16] ensures

‖bWν (x,D)‖ � Cmn, ‖(I − l0 W

n (x,D))bWν (x,D)‖ = O(hNn ) (6.2)

for every N ∈ N.We introduce the following formal notation:

Y (τ, B) := e−iτP BeiτP . (6.3)

Let {Yν}ν∈V be a family of bounded operators and let m ∈ Mc(g). We writeYν ∈ Y(m) if and only if there exist N ∈ N, C0 > 0, the weights m(k, k′) ∈Mc(g), the symbols bk,k′,ν ∈ Sn(m(k, k

′), g) and functions sk,k′ : [0; 1]N → R,sk,ν : [0; 1]N → C, satisfying

N∏k′=1

m(k, k′) � m, |sk,k′(w)| � C0, |sk,ν(w)| � C0〈hnpnt〉C0,

for k, k′ = 1, . . . , N and

Yν =N∑k=1

∫[0; 1]N

dwsk,ν(w)Y (sk,1(w)τ, Bk,1,ν) . . . Y (sk,N(w)τ, Bk,N,ν) + Rν, (6.4)

where Bk,k′,ν = bWk,k′,ν(x,D) and the family {Rν}ν∈V is negligible.

Taking sk,ν(w) = sk,k′(w) = 1, we can forget the integration with respect to w,hence

Y (τ, Bk,1,ν) . . . Y (τ, Bk,N,ν) ∈ Y(m)

and, more generally,

Yν ∈ Y(m), Yν ∈ Y(m) �⇒ YνYν ∈ Y(mm), (6.5)

Yν ∈ Y(m) �⇒ ‖Yν‖ � Cmn〈hnpnt〉C. (6.6)

6.2. REFORMULATION OF PROPOSITION 5.3

In Section 7 we will prove


PROPOSITION 6.1. Assume q0,ν ∈ S(m), Y0,ν ∈ Y(1) and N0 ∈ N. Then one canfind k0 ∈ N

∗ and C(N0) > 0 such that

tJν(q0, Y0) =∑

1�k�k0

(Jν(qk, Yk) + tJν(q−k , Y−k))+

+ O(hN0n 〈hnpnt〉C(N0)

)(6.7)

holds with certain symbols q±k,ν and operators Y±k,ν satisfying

qk,ν ∈ S(m/p), q−k,ν ∈ S(mhσ ), Y±k,ν ∈ Y(1) (6.8)

for k = 1, . . . , k0.

It is easy to see that Proposition 6.1 implies Proposition 5.3. Indeed, first of allwe note that the assertion of Proposition 6.1 can be applied to express tJν(q−k , Y−k),k = 1, . . . , k0 and iterating this procedure N times, we find the expression oftJν(q0, Y0) in the form (6.7) with new symbols q−k,ν ∈ S(mhNσ ), k = 1, . . . , kN .Thus, for N = N(N0) large enough, all terms tJν(q−k , Y−k ), k = 1, . . . , kN , be-come O(hN0

n 〈hnpnt〉C(N0)), i.e. the assertion of Proposition 6.1 holds with q−k = 0for k � 1. This proves Proposition 5.3 in the case k = 1 and it is clear that thegeneral case follows after k iterations.

In the remaining part of this section, we describe the properties of Yν ∈ Y(m)needed in the proof of Proposition 6.1. More precisely, we consider the commutatorof Yν with the operator of multiplication by the j th coordinate, denoted by xj .

LEMMA 6.2. Assume Yν ∈ Y(m). Then there exist

Y+ν ∈ Y(〈ξ 〉−1m), Y−

ν ∈ Y(〈ξ 〉−1phσm) (6.9)

such that [Yν, xj ] = Y+ν + τY−

ν .Proof. Let Bk,k′,ν = bW

k,k′,ν(x,D) with bk,k′,ν ∈ Sn(m(k, k′), g). If we know that

[Y (τ, Bk,k′,ν), xj ] = Y+k,k′,ν + τY−

k,k′,ν (6.10)

holds with

Y+k,k′,ν ∈ Y(〈ξ 〉−1m(k, k′)), Y−

k,k′,ν ∈ Y(〈ξ 〉−1phσm(k, k′)),

then succesively commuting xj with Y (sk,k′(w)τ, Bk,k′,ν), k′ = 1, . . . , N , we ob-tain easily the general statement of Lemma 6.2.

To begin we write

[Y (τ, Bk,k′,ν), xj ] = e−iτP [Bk,k′,ν, Y (−τ, xj )]eiτP (6.11)

and denote

Pj := [iP, xj ] = ∂ξj pW (x,D) = ∂ξj p0(x,D). (6.12)

170 LECH ZIELINSKI

Then we can write the Taylor formula

Y (−τ, xj ) = xj − τ∂τY (0, xj ) + τ 2∫ 1

0ds(1 − s)∂2

τ Y (−sτ, xj )

= xj + τPj + τe−iτP Y (τ, [iτP, Pj ])eiτP , (6.13)

where

Y (τ, B) = τ

∫ 1

0ds(1 − s)Y ((−s − 1)τ, B).

Using (6.13) we can express the commutator (6.11) in the form

Y (τ, [Bk,k′,ν, xj ]) + τY (τ, [Bk,k′,ν, Pj ]) + τ [Y (τ, Bk,k′,ν), Y (τ, [iτP, Pj ]) (6.14)

and since [Bk,k′,ν, xj ] = ∂ξj bWk,k′,ν(x,D), it is clear that the first term of (6.14) is

in Y(〈ξ 〉−1m(k, k′)). Then Y (τ, [Bk,k′,ν, Pj ]) ∈ Y(〈ξ 〉−1phσm(k, k′)) is a conse-quence of

bk,k′,ν � ∂ξj p − ∂ξj p � bk,k′,ν ∈ Sn(〈ξ 〉−1phσm(k, k′), g),

which follows from (3.23) due to

∂xj ′bk,k′,ν ∈ Sn(〈ξ 〉hσm(k, k′), g), ∂ξj ′ ∂ξj p ∈ S(〈ξ 〉−2p, g),

∂ξj ′bk,k′,ν ∈ Sn(〈ξ 〉−1m(k, k′), g), ∂xj ′ ∂ξj p ∈ S(ph1+σ , g) ⊂ S(phσ , g)

(we use hp � p in the last inclusion). Moreover, using

∂xj ′p ∈ S(〈ξ 〉h1+σp, g), ∂ξj ′p ∈ S(〈ξ 〉−1p, g),

we find that (3.23) ensures

p � ∂ξj p − ∂ξj p � p ∈ S(〈ξ 〉−1ph1+σp, g). (6.15)

Introducing l0n ∈ S(1, g) such that (I −L0n)Bk,k′,ν is negligible with L0

n = l0n(x,D)

and supp l0n ⊂ Bn(c0) with c0 < c, we can write

Y (τ, Bk,k′,ν)Y (τ, [Pj , iτP ])= Y (τ, Bk,k′,ν)Y (τ, L

0n[Pj , iτP ]) + Rk,k′,ν (6.16)

with {Rk,k′,ν}ν∈V negligible. However, using

〈hnpnτ 〉−1|τ |l0n ∈ Sn(h−1p−1, g)

and (6.15), we find pj,ν ∈ Sn(〈ξ 〉−1phσ , g) such that

〈hnpnτ 〉−1τL0n[Pj , iP ] = pW

j,ν(x,D),

hence Y (τ, L0n[Pj , iτP ]) ∈ Y(〈ξ 〉−1phσ ). Therefore the right-hand side of (6.16)

belongs to Y(〈ξ 〉−1phσm(k, k′)) and Y(τ, [Pj , iτP ])Y (τ, Bk,k′,ν) belongs to thesame class, i.e. (6.14) gives the desired decomposition (6.10). ✷


COROLLARY 6.3. Let Y0,ν ∈ Y(1). If Pj = [iP, xj ], then one has

τPje−iτP Y0,ν = [e−iτP Y0,ν, xj ] + e−iτP (Y+0,ν + τY−

0,ν) (6.17)

with some Y+0,ν ∈ Y(〈ξ 〉−1) and Y−

0,ν ∈ Y(〈ξ 〉−1phσ ).Proof. Indeed, using (6.13) to express [e−iτP , xj ] and applying Lemma 6.2 with

Yν = Y0,ν we can write

[e−iτP Y0,ν, xj ]= [e−iτP , xj ]Y0,ν + e−iτP [Y0,ν, xj ]= −τPje−iτP Y0,ν − τe−iτP Y (τ, [Pj , iτP ])Y0,ν + e−iτP (Y+

ν + τY−ν ).

It remains to remark that the reasoning of the proof of Lemma 6.2 ensures the factthat Y+

0,ν = Y+ν and Y−

0,ν = −Y(τ, [Pj , iτP ])Y0,ν + Y−ν belong to the indicated

classes. ✷

7. End of the Proof of Theorem 2.1

Throughout this section, we use the following notation:

Qν = Op(ei(τ−t )pqν) with qν ∈ S(〈ξ 〉m/p). (7.1)

We adopt the convention that the symbol (x, ξ) → p(x, ξ) can be considered as(x, ξ, y) → p(x, ξ), allowing us to define qν∂ξj p ∈ S(m) by the formula

(qν∂ξj p)(x, ξ, y) = qν(x, ξ, y)∂ξj p(x, ξ). (7.2)

LEMMA 7.1. If Qν and qν∂ξj p are as in (7.1), (7.2), then

[Qν, xj ] = (t − τ)Op(ei(τ−t )pqν∂ξj p) + Op(ei(τ−t )pi∂ξj qν). (7.3)

Proof. Since the integral kernel of [Qν, xj ] is

(x, y) &→ (2π)−d

∫(yj − xj )e

i(x−y)ξ+i(τ−t )p(x,ξ)qν(x, ξ, y) dξ (7.4)

and

(yj − xj )ei(x−y)ξ = −i∂ξj e

i(x−y)ξ ,

the integration by parts allows us to write (7.4) in the form

(x, y) &→ (2π)−d

∫ei(x−y)ξ+i(τ−t )p(x,ξ)((t − τ)∂ξj pqν + i∂ξj qν)(x, ξ, y) dξ,

which gives (7.3). ✷

172 LECH ZIELINSKI

The computation of the composition kernel gives

Op(ei(τ−t )pqν)bν(x,D)∗ = Op(ei(τ−t )p(qν • bν)) (7.5)

with

(qν • bν)(x, ξ, y) = (2π)−d

∫ei(y−y)(ξ−ξ)qν(x, ξ, y)bν(y, ξ )

and the usual Taylor development with integrations by parts give

qν • bν = qν•N ′bν + rN ′,ν, (7.6)

with

(qν•N ′bν)(x, ξ, y) =∑

|α|<N ′(−i)|α|∂αy qν(x, ξ, y)∂

αξ bν(y, ξ)/α!,

rN ′,ν =∑

|α|=N ′(qα,ν(z) • ∂αξ bν)|z=y, (7.7)

qα,ν(z)(x, ξ, y) =∫ 1

0∂αy qν(x, ξ, z + s(y − z))

iN′N ′ ds

α! .

Then, for an arbitrary N ∈ N, we can find N ′ = N ′(N) such that the family ofoperators RN ′,ν = Op(ei(τ−t )prN ′,ν) satisfies

‖RN ′,ν‖ � CNhNn 〈hnpnt〉C(N). (7.8)

Denoting (qνbBν)(x, ξ, y) = qν(x, ξ, y)bν(y, ξ), we can write

qν ∈ S(m), bν ∈ S(m′, g) ⇒ qν•N ′bν ∈ S(mm′),qν•N ′bν − qνb

Bν ∈ S(mm′hσ ).

If, in particular, qν ∈ S(〈ξ 〉m/p), ∂ξj p0 ∈ S(〈ξ 〉−1p, g), then

QνPj = Op(ei(τ−t )p(qν • ∂ξj p0)),

qν•N ′∂ξj p0 ∈ S(m),

qν•N ′∂ξj p0 − qν∂ξj p0B ∈ S(hσm).

Since (3.20) ensures ∂ξj p − ∂ξj p0 ∈ S(〈ξ 〉−1h1+σp, g) ⊂ S(〈ξ 〉−1hσ p, g),

q−N ′,ν := qν•N ′∂ξj p0 − qν∂ξj p

B ∈ S(hσm). (7.9)

For j = 1, . . . , d, we introduce

p(j),ν(x, ξ, y) :=(

1 − τ

t

)∂ξj p(x, ξ) + τ

t∂ξj p(y, ξ). (7.10)


PROPOSITION 7.2. Let qν ∈ S(〈ξ 〉m/p), Y0,ν ∈ Y(1) and N0 ∈ N. Then we canfind C(N0) > 0 such that

tJν(qp(j), Y0) =∑

1�k�2

(Jν(qk, Yk) + tJν(q−k, Y−k)) + O(hN0n 〈hnpnt〉C(N0)

), (7.11)

where for k = 1 and 2 we have

qk,ν ∈ S(m/p), q−k,ν ∈ S(mhσ ), Yk,ν ∈ Y(1), Y−k,ν ∈ Y(1). (7.12)

Proof. By definition of p(j),ν , we have

t Op(ei(τ−t )pqνp(j),ν)

= (t − τ)Op(ei(τ−t )pqν∂ξj p) + τOp(ei(τ−t )pqν∂ξj pB) (7.13)

and applying (7.3), (7.9), we can write the above expression as

[Qν, xj ] + QντPj + Op(ei(τ−t )p(τq−N ′,ν − i∂ξj qν)) + τRN ′,ν

with RN ′,ν satisfying (7.8). Thus tJν(qp(j), Y0) can be written as

tr[Qν, xj ]e−iτP Y0,ν + trQντPje−iτP Y0,ν ++ Jν(−i∂ξj q, Y0) + τJν(q

−N ′ , Y0) + O(hN0

n 〈hnpnt〉C(N0)). (7.14)

Due to Corollary 6.3, the sum of two first terms in (7.14) equals

tr[Qν, xj ]e−iτP Y0,ν + trQν[e−iτP Y0,ν, xj ] + Jν(q, Y+0 ) + τJν(q, Y

−0 )

= tr[Qνe−iτP Y0,ν, xj ] + Jν(q, Y+0 ) + τJν(q, Y

−0 )

= Jν(q, Y+0 ) + τJν(q, Y

−0 ). (7.15)

Thus, we obtain (7.11) with

q1,ν = i∂ξj qν, q−1,ν = τ

tq−N ′,ν, Y1,ν = Y−1,ν = Y0,ν,

q2,ν = 〈ξ (k+n )〉−1

qν, Y2,ν = 〈ξ (k+n )〉Y+

0,ν,

q−2,ν = 〈ξ (k+n )〉−1

pnhσnqν, Y−2,ν = 〈ξ (k+

n )〉p−1n h−σ

n Y−0,ν,

where, according to our convention, pn = pn(x(k−n ), ξ (k

+n )). ✷

Proof of Proposition 6.1. Let χs , χs be as in the proof of Corollary 4.3, then

q0,ν ∈ S(m, g) �⇒ qν := q0,νχsh−1n p−1

n ∈ S(m/p, g)

and

tJν(q0χs, Y0) = Jν(q, Y ) with Yν = hnpntY0,ν ∈ Y(1).

Therefore, it suffices to prove the statement of Proposition 6.1 with q0,νχs insteadof q0,ν .

174 LECH ZIELINSKI

Further on, we assume s = 1 and note that for (x, ξ) ∈ supp χ1 we have

〈ξ 〉−1p(x, ξ)/2 � |∇ξp(x, ξ)| � 2〈ξ 〉−1p(x, ξ) (7.16)

and, due to (2.10), for (x, ξ, y) ∈ supp q0,νχ1, we have

|∇ξp(x, ξ)| −(

d∑j=1

p2(j),ν

)1/2

� |∇ξp(x, ξ) − ∇ξp(y, ξ)|� 〈ξ 〉−1p(x, ξ)/4 � |∇ξp(x, ξ)|/2,

which implies(d∑

j=1

p2(j),ν

)1/2

� |∇ξp(x, ξ)|/2 � 〈ξ 〉−1p(x, ξ)/4. (7.17)

Using (7.16), (7.17), we can write q0,ν χ1 = ∑dj=1 q(j),νp(j),ν with

q(j),ν = q0,νχ1p(j),ν

(d∑

j=1

p2(j),ν

)−1

∈ S(〈ξ 〉m/p, g)

and we complete the proof applying Proposition 7.2 with qν = q(j),ν . ✷

8. Proof of Theorem 2.2

8.1. THE FAMILY OF MOLLIFYING FUNCTIONS

Let γ, γ ∈ C∞0 ({x ∈ R

d : |x| < c}) be such that∫γ (x) dx =

∫γ (x) dx = 1, γ � 0

and γ (x) � c0 > 0 for x ∈ supp γ . For s > 0, we denote

γs(x) := s−dγ (x/s), γs(x) := s−d γ (x/s).

We note that∫γs(x) dx = ∫

γs(x) dx = 1 and

|∂αx,sγs(x)| � Cαs−|α|γs(x). (8.1)

8.1.1. Mollifying v

Let 0 < ε � (ρ − ρ ′)/2, r ′/r < δ < 1 and define

sk(ξ′, ξ ) = v(x(k))ε〈ξ ′, ξ 〉−δ

, (8.2)


where

k ∈ N, ξ ′, ξ ∈ Rd and 〈ξ ′, ξ 〉 := (1 + |ξ ′|2 + |ξ |2)1/2.

It is easy to check that

|∂α′ξ ′ ∂αξ sk(ξ

′, ξ )| � Cα′,αsk(ξ′, ξ )〈ξ, ξ ′〉−|α′|−|α|

. (8.3)

Using the partition of unity of Section 3, we set

vk := vθ−k and vk(ξ

′, x, ξ) := (vk ∗ γsk(ξ ′,ξ ))(x). (8.4)

Similarly as (2.25), we check that |x − y| � cv(x)ε implies

|v(x) − v(y)| � C max{|x − y|r , 1 + |x − y|}v(x)1−ρ

� 2C|x − y|rv(x)1−ρ+(1−r)ε. (8.5)

Using (8.5) to estimate

vk(ξ′, x, ξ) − vk(x) =

∫(vk(y) − vk(x))γsk(ξ ′,ξ )(x − y) dy,

we find

|vk(ξ ′, x, ξ) − vk(x)|� Cv(x(k))1−ρ+(1−r)ε

∫|x − y|r |γsk(ξ ′,ξ )(x − y)| dy

= Cv(x(k))1−ρ+(1−r)εsk(ξ′, ξ )r

∫|y|r |γ (y)| dy

� C ′v(x)1−ρ+ε〈ξ ′, ξ 〉−δr . (8.6)

If |α′| + |α| + |β| � 1, then∫∂α

′ξ ′ ∂αξ ∂

βx γsk(ξ ′,ξ )(x − y) dy = 0 and

∂α′

ξ ′ ∂αξ ∂βx vk(ξ

′, x, ξ) =∫(vk(y) − vk(x))∂

α′ξ ′ ∂αξ ∂

βx γsk(ξ ′,ξ )(x − y) dy. (8.7)

However, using (8.1) and (8.3), we can find

|∂α′ξ ′ ∂αξ ∂

βx γsk(ξ ′,ξ )(x − y)| � Cα′,α,β γsk(ξ ′,ξ )(x − y)sk(ξ

′, ξ )−|β|〈ξ ′, ξ 〉−|α′|−|α|

and, estimating the integral (8.7), we find

|∂α′ξ ′ ∂αξ ∂

βx vk(ξ

′, x, ξ)| � Cα′,α,βv(x)1−ρ+εsk(ξ

′, ξ )r−|β|〈ξ ′, ξ 〉−|α′|−|α|. (8.8)

8.2. ESTIMATES OF PSEUDODIFFERENTIAL OPERATORS

We fix σ > 0 small enough to ensure

(1 + σ )r ′ < rδ, (1 + σ )ρ ′ � ρ − ε, σ r ′ � 1 − δ, σρ ′ � ε.

176 LECH ZIELINSKI

These conditions on σ ensure

v(x)ε−ρ〈ξ 〉−rδ � v(x)−(1+σ)ρ′ 〈ξ 〉−(1+σ)r ′ = h(x, ξ)1+σ , (8.9)

v(x)−ε〈ξ 〉δ−1 � v(x)−σρ′ 〈ξ 〉−σr ′ = h(x, ξ)σ . (8.10)

We define V = ∑∞k=0 Vk, where Vk has the kernel given by the oscillatory integral

(x, y) → (2π)−2d∫

ei(x−x ′)ξ ′+i(x ′−y)ξ vk(ξ′, x, ξ) dξ ′ dx′ dξ.

Using Theorem 2.1 of Kumano-Go and Nagase [20], we note that the estimates∣∣∂α′ξ ′ ∂αξ

(〈ξ ′〉s/2(vk(x) − vk(ξ

′, x, ξ))〈ξ 〉s/2)∣∣� Cα′,αv(x(k))

1−ρ+ε〈ξ ′〉(s−rδ)/2−|α′|〈ξ 〉(s−rδ)/2−|α|

with s = (1 + σ )r ′ < rδ ensure

‖〈D〉(1+σ)r ′/2(Vk − Vk)〈D〉(1+σ)r ′/2‖ � Cv(x(k))1−ρ+ε, (8.11)

where Vk denotes the operator of multiplication by vk. The estimate (8.11) can bewritten as the inequality (in the sense of quadratic forms)

±(Vk − Vk) � Cv(x(k))1−ρ+ε〈D〉−(1+σ)r ′. (8.12)

Let 5−k be as in Section 5 and consider 5−

k Vk5−k = v0

k (x,D). Then 5−k Vk5

−k = Vk

and standard symbol expansions (cf. [19, 20]) give the approximation of v0k by∑

|α|�N

i|α|∂αx ∂αξ ′ vk(ξ

′, x, ξ)|ξ ′=ξ /α!, (8.13)

hence (8.6), (8.8), (8.9), (8.10) imply

|v0k (x, ξ) − vk(x)| � Cv(x)1−ρ+ε〈ξ 〉−rδ � Cv(x)h(x, ξ)1+σ , (8.14)

|∂βx ∂αξ v0k (x, ξ)| � Cα,βv(x)

1−ρ+ε〈ξ 〉−rδ(v(x)−ε〈ξ 〉δ−1)|β|〈ξ 〉|β|−|α|

� C ′α,βv(x)h(x, ξ)

1+σ(1+|β|)〈ξ 〉|β|−|α|, (8.15)

if |α| + |β| � 1.LetAj,j ′,k denote the operator of multiplication by the function aj,j ′,k := θ−

k aj,j ′and similarly as before, let

aj,j ′,k(ξ′, x, ξ) := (aj,j ′,k ∗ γsk(ξ ′,ξ ))(x). (8.16)

Reasoning as before, we find the estimates

|aj,j ′,k(x) − aj,j ′,k(ξ′, x, ξ)| � Cv(x)ε−ρ〈ξ ′, ξ 〉−rδ

, (8.17)

|∂α′ξ ′ ∂αξ ∂

βx aj,j ′,k(ξ

′, x, ξ)| � Cα′,α,βv(x)ε(1−|β|)−ρ〈ξ ′, ξ 〉δ(|β|−r)−|α′|−|α|

, (8.18)


if |α′| + |α| + |β| � 1. Using [20] similarly as before, we find

‖〈D〉(1+σ)r ′/2(Aj,j ′,k − Aj,j ′,k)〈D〉(1+σ)r ′/2‖ � Cv(x(k))ε−ρ, (8.19)

if the operator Aj,j ′,k has the kernel

(x, y) → (2π)−d

∫ei(x−x ′)ξ ′+i(x ′−y)ξ aj,j ′,k(ξ

′, x, ξ) dξ ′ dx′ dξ

and (8.19) implies∣∣∣∣ ∑1�j,j ′�d

( (Aj,j ′,k − Aj,j ′,k)Djϕ,Dj ′ϕ)

∣∣∣∣� Cv(x(k))ε−ρ(〈D〉2−(σ+r ′)ϕ, ϕ). (8.20)

Introducing Ak,0 = ∑1�j,j ′�d Dj ′Aj,j ′,kDj , we find 5−

k Ak,05−k = ak,0(x,D) with

ak,0 satisfying

|∂αξ (ak,0 − a0)(x, ξ)| � Cαv(x)ε−ρ〈ξ 〉2−δr−|α|, (8.21)

|∂αξ ∂βx ak,0(x, ξ)| � Cα,βh(x, ξ)1+σ |β|〈ξ 〉2+|β|−|α|, (8.22)

where a0 is given by (1.4), α ∈ Nd , β ∈ N

d \ {0}.If C > 0 is large enough, then (8.12), (8.20) ensure (2.1) with

P± =∞∑k=1

5−k (A0,k + Vk ± CRk)5

−k ,

Rk = v(x(k))1−(1+σ)ρ′〈D〉−(1+σ)r ′ + v(x(k))−(1+σ)ρ′ 〈D〉2−(1+σ)r ′

and R = ∑∞k=1 5

−k Rk5

−k = rW (x,D) holds with r ∈ S(ah1+σ , g), hence p = p±

satisfy the hypotheses of Theorem 2.1 due to Lemma 2.3 and remark at the end ofSection 3.

9. Appendix

LEMMA 9.1. (a) LetQν = (e−itpqν)(x,D) with qν ∈ S(m, g) such that supp qν ⊂Bn(c). If C > 0 is large enough, then

‖Qν‖tr � mC,ν := Cmnh−Cn 〈hnpnt〉C. (9.1)

(b) If Qν = Op(e−itpqν) with qν ∈ S(m), then the estimate (9.1) still holds.Proof. For k ∈ N let Fk(x, ξ) = 〈x〉2k〈ξ 〉2k. If k > d, then Fk(x,D)−1 is of

trace class and

‖Qν‖tr � ‖Fk(x,D)−1‖tr ‖Fk(x,D)Qν‖.

178 LECH ZIELINSKI

A direct calculus gives Fk(x,D)Qν = Op(e−itp)qk,ν with |qk,ν | � Ckm|t|2kp2kFk

and for C > 0 large enough, we obtain |qk,ν | � mC,ν . Using the same kernelexpression (5.6) in cases (a) and (b), we complete the proof due to Schwarz lemma

‖Op(e−itpqk,ν)‖ �(

suppy

∫|Kν(q, x, y)| dx suppx

∫|Kν(q, x, y)| dy

)1/2

� mC,ν vol(Bn(c)) � C ′mC,νh−C ′n . ✷

Proof of Proposition 1.2. Our aim is to estimate the volume of

G(λ) = {(x, ξ) ∈ R2d : a(1 − C0h)(x, ξ) < λ < a(1 + C0h)(x, ξ),

|x| + |ξ | > C1}.We assume C1 > 0 large enough to ensure

λ/2 � a(x, ξ) � 2λ for (x, ξ) ∈ G(λ).

Case 1. We consider (x, ξ) ∈ G(λ) satisfying v(x) � a0(x, ξ). Then

λ/2 � a(x, ξ) � 2a0(x, ξ) � C|ξ |2 (9.2)

ensures

(ah)(x, ξ) � 2a0(x, ξ)〈ξ 〉−r ′v(x)−ρ′ � C ′a0(x, ξ)λ

−r ′/2v(x)−ρ′(9.3)

and (x, ξ) ∈ G(λ) implies

a0(x, ξ)(1 − C ′0λ

−r ′/2v(x)−ρ′)

� λ − v(x) � a0(x, ξ)(1 + C ′0λ

−r ′/2v(x)−ρ′). (9.4)

If λ � λ0 with λ0 > 0 large enough, then (9.4) implies λ − v(x) � 0 and

λ−(x) � a0(x, ξ) � λ+(x) (9.5)

with

λ±(x) = (λ − v(x))(1 ± 2C ′0λ

−r ′/2v(x)−ρ′). (9.6)

Indeed, if λ±(x) are given by (9.6), then

λ−(x) � (λ − v(x))(1 + C ′0λ

−r ′/2v(x)−ρ′)−1

� (λ − v(x))(1 − C ′0λ

−r ′/2v(x)−ρ′)−1 � λ+(x),

implying (9.5).Since for d � 2 we have∫

λ−(x)<a0(x,ξ)<λ+(x)dξ = ω(x)(λ+(x)d/2 − λ−(x)d/2)

� C(λ+(x) − λ−(x))λ+(x)d/2−1 (9.7)


and λ±(x) � Cλ, we have∫λ−(x)<a0(x,ξ)<λ+(x)

dξ � C ′v(x)−ρ′λd/2−r ′/2. (9.8)

Case 2. We note that for a given C > 0 we have∫|ξ |2+r′�Cλ1−ρ′

dξ = C ′λ1−ρ′2+r′ d (9.9)

and

d � 3 �⇒ 1 − ρ ′

2 + r ′ d � d − r ′

2− ρ ′.

Case 3. We consider (x, ξ) ∈ G(λ) satisfying a0(x, ξ) � v(x). Then

λ/2 � a(x, ξ) � 2v(x) �⇒ (ah)(x, ξ) � Cλ1−ρ′a0(x, ξ)

−r ′/2, (9.10)

hence (x, ξ) ∈ G(λ) implies

a0(x, ξ) − C1λ1−ρ′

a0(x, ξ)−r ′/2

< λ − v(x) < a0(x, ξ) + C1λ1−ρ′

a0(x, ξ)−r ′/2. (9.11)

Due to (9.9), it remains to consider the region where

a0(x, ξ)1+r ′/2 � c|ξ |2+r ′ � 3C1λ

1−ρ′(9.12)

and the condition (9.12) ensures

λ − v(x) > a0(x, ξ) − C1λ1−ρ′

a0(x, ξ)−r ′/2 � 2C1λ

1−ρ′a0(x, ξ)

−r ′/2,

λ − v(x) < a0(x, ξ) + C1λ1−ρ′

a0(x, ξ)−r ′/2 � a0(x, ξ) + (λ − v(x))/2,

hence 2a0(x, ξ) � λ − v(x).Due to the last inequality, (9.11) implies (9.5) with

λ±(x) = λ − v(x) ± 2C1λ1−ρ′

(λ − v(x))−r ′/2. (9.13)

Let us check that

λ+(x) � C(λ − v(x)). (9.14)

Indeed, using once more (9.10) and (9.11), we can write

λ − v(x) > a0(x, ξ) − C1λ1−ρ′

a0(x, ξ)−r ′/2 � a0(x, ξ)/2,

hence

12C1λ

1−ρ′> a0(x, ξ)

1+r ′/2 �(2(λ − v(x))

)1+r ′/2

and (9.14) follows.

180 LECH ZIELINSKI

Using (9.7) with (9.13) we find∫λ−(x)<a0(x,ξ)<λ+(x)

dξ � Cλ1−ρ′(λ − v(x))−r ′/2λ+(x)d/2−1

� C ′λ1−ρ′(λ − v(x))(d−r ′)/2−1

� C ′λ(d−r ′)/2−ρ′.

To complete the proof of Proposition 1.2, it remains to perform the integration withrespect to x in the region v(x) � 2λ. ✷

9.1. CASES d = 1 AND d = 2

Using (9.9) to estimate the integral in the left-hand side of (1.15) in the regionv(x) < λ, |ξ |2+r ′ � Cλ1−ρ′

, we obtain an additional term V(λ)O(λ(1−ρ′)d/(2+r ′)) inthe right-hand side of (1.15). However, sometimes we can obtain a better estimateif there exists κ ∈ [0; 1] such that

1 � τ � λ/2 �⇒∫λ−τ<v(x)<λ+τ

dx � CτV(λ)λ−κ . (9.15)

Indeed, using (9.15) instead of (9.9), we find the error estimate

CV(λ)λ1−ρ′−κ

∫|ξ |2+r′�Cλ1−ρ′

〈ξ 〉−r ′/2 dξ � CV(λ)λ1−ρ′−κ+ 1−ρ′

2+r′ (d− r′2 ).

Acknowledgement

This paper would not be written without aid of Janina Dawczynska and it is dedi-cated to her memory.

References

1. Birman, M. and Solomyak, M.: Asymptotic behaviour of spectrum of differential equations,J. Soviet. Math. 12 (1979), 247–282.

2. Boimatov, K.: Spectral asymptotics of pseudodifferential operators, Soviet Math. Dokl. 42(2)(1990), 196–200.

3. Buzano, E.: Some remarks on the Weyl asymptotics by the approximate spectral projectionmethod, Boll. Un. Mat. Ital. Sez. B Artic. Ric. Mat. 8 (2000), 775–792.

4. Dencker, N.: The Weyl calculus with locally temperate metrics and weights, Ark. Mat. 24(1986), 59–79.

5. Edmunds, D. E. and Evans, W. D.: On the distribution of eigenvalues of Schrödinger operators,Arch. Rational Mech. Anal. 89 (1985), 135–167.

6. Edmunds, D. E. and Evans, W. D.: Spectral Theory and Differential Operators, Oxford Math.Monogr., Oxford, 1989.


7. Faris, W. G.: Self-adjoint Operators, Lecture Notes in Math. 433, Springer-Verlag, New York,1975.

8. Feigin, V. I.: The asymptotic distribution of the eigenvalues of pseudodifferential operators inRn, Math. USSR-Sb. 28 (1976), 533–552.

9. Feigin, V. I.: Sharp estimates of the remainder in the spectral asymptotics for pseudodifferentialoperators in Rn, Functional Anal. Appl. 16 (1982), 88–89.

10. Fleckinger, J. and Lapidus, M.: Remainder estimates for the asymptotics of elliptic eigenvalueproblems with indefinite weights, Arch. Rational Mech. Anal. 98 (1987), 329–356.

11. Fleckinger, J. and Lapidus, M.: Schrödinger operators with indefinite weight functions:asymptotics of eigenvalues with remainder estimates, Differential Integral Equations 7 (1994),1389–1418.

12. Guillemin, V. and Sternberg, S.: Some problems in integral geometry and some relatedproblems in microlocal analysis, Amer. J. Math. 101 (1979), 915–955.

13. Helffer, B.: Théorie spectrale pour des opérateurs globalement elliptiques, Astérisque 112(1984).

14. Helffer, B. and Robert, D.: Propriétés asymptotiques du spectre d’opérateurs pseudo-différentiels sur Rn, Comm. Partial Differential Equations 7 (1982), 795–881.

15. Hörmander, L.: On the asymptotic distribution of the pseudodifferential operators in Rn, Ark.Mat. 17 (1979), 297–313.

16. Hörmander, L.: The Analysis of Linear Partial Differential Operators, vols 1–4, Springer-Verlag, New York, 1983, 1985.

17. Ivrii, V.: Microlocal Analysis and Precise Spectral Asymptotics, Springer-Verlag, Berlin, 1998.18. Ivrii, V.: Sharp spectral asymptotics for operators with irregular coefficients, Internat. Math.

Res. Notices (2000), 1155–1166.19. Kumano-Go, H.: Pseudodifferential Operators, MIT Press, Cambridge, MA, 1981.20. Kumano-Go, H. and Nagase, M.: Pseudodifferential operators with non-regular symbols and

applications, Funkcial. Ekvac. 21 (1978), 151–192.21. Levendorskii, S. Z.: Asymptotic Distribution of Eigenvalues of Differential Operators, Math.

Appl., Kluwer Acad. Publ., Dordrecht, 1990.22. Métivier, G.: Valeurs propres des problèmes aux limites irréguliers, Bull. Soc. Math. France

Mem. 51–52 (1977), 125–219.23. Miyazaki, Y.: A sharp asymptotic remainder estimate for the eigenvalues of operators associ-

ated with strongly elliptic sesquilinear forms, Japan. J. Math. 15 (1989), 65–97.24. Miyazaki, Y.: The eigenvalue distribution of elliptic operators with Hölder continuous coeffi-

cients, Osaka J. Math. 28 (1991), 935–973; Part 2, Osaka J. Math. 30 (1993), 267–302.25. Mohamed, A.: Comportement asymptotique avec estimation du reste, des valeurs propres d’une

classe d’opérateurs pseudo-différentiels sur Rn, Math. Nachr. 140 (1989), 127–186.26. Reed, M. and Simon, B.: Methods of Modern Mathematical Physics, vols I–IV, Academic Press,

New York, 1972, 1975, 1979.27. Robert, D.: Propriétés spectrales d’opérateurs pseudo-différentiels, Comm. Partial Differential

Equations 3 (1978), 755–826.28. Rozenblyum, G. V.: Asymptotics of the eigenvalues of the Schrödinger operator, Math. USSR-

Sb. 22 (1974), 349–371.29. Shubin, M. A. and Tulovskii, V. A.: On the asymptotic distribution of eigenvalues of

pseudodifferential operators in Rn, Math. USSR-Sb. 21 (1973), 565–573.30. Tamura, H.: Asymptotic formula with remainder estimates for eigenvalues of Schrödinger

operators, Comm. Partial Differential Equations 7 (1982), 1–54.31. Tamura, H.: Asymptotic formula with sharp remainder estimates for eigenvalues of elliptic

operators of second order, Duke Math. J. 49 (1982), 87–119.32. Zielinski, L.: Asymptotic behaviour of eigenvalues of differential operators with irregular

coefficients on a compact manifold, C.R. Acad. Sci. Paris Sér. I Math. 310 (1990), 563–568.

182 LECH ZIELINSKI

33. Zielinski, L.: Asymptotic distribution of eigenvalues for elliptic boundary value problems,Asymptotic Anal. 16 (1998), 181–201.

34. Zielinski, L.: Asymptotic distribution of eigenvalues of some elliptic operators with intermedi-ate remainder estimates, Asymptotic Anal. 17 (1998), 93–120.

35. Zielinski, L.: Sharp spectral asymptotics and Weyl formula for elliptic operators with non-smooth coefficients, Math. Phys. Anal. Geom. 2 (1999), 291–321; Part 2, Colloq. Math. 92(2002), 1–18.


183

Heat Kernel Expansions on the Integers

F. ALBERTO GRÜNBAUM and PLAMEN ILIEVDepartment of Mathematics, University of California, Berkeley, CA 94720-3840, U.S.A.e-mail: {grunbaum, iliev}@math.berkeley.edu

(Received: 27 December 2001; in final form: 21 April 2002)

Abstract. In the case of the heat equation ut = uxx +V u on the real line, there are some remarkablepotentials V for which the asymptotic expansion of the fundamental solution becomes a finite sumand gives an exact formula.

We show that a similar phenomenon holds when one replaces the real line by the integers. In thiscase the second derivative is replaced by the second difference operator L0. We show if L denotes theresult of applying a finite number of Darboux transformations to L0 then the fundamental solutionof ut = Lu is given by a finite sum of terms involving the Bessel function I of imaginary argument.

Mathematics Subject Classifications (2000): 35Q58, 37K20, 58J72.

Key words: heat kernel expansions, Toda lattice hierarchy, Darboux transformations.

1. Heat Kernel Expansions

The subject of heat kernel expansions on Riemannian manifolds with or withoutboundaries cuts across a number of branches of mathematics and serves as aninteresting playground for a whole array of interactions with physics.

It suffices to mention, for instance, the work of Kac [18] on the issue of recov-ering the shape of a drum from its pure tones, as well as the suggestion by McKeanand Singer [20] that one should be able to find a ‘heat equation proof’ of the indextheorem. For an updated account, see [8, 23].

In the simpler case of the whole real line, the fundamental solution of theequation

ut = uxx + V (x)u, u(x, 0) = δy(x)

admits an asymptotic expansion valid for small t and x close to y, in the form

u(x, y, t) ∼ e− (x−y)2

4t√4πt

(1 +

∞∑n=1

Hn(x, y)tn

).

When V is taken to be a potential such that L = (d/dx)2 + V belongs to a rankone bispectral ring, then something remarkable happens, namely this expansiongives rise to an exact formula consisting of a finite number of terms and valid for

184 F. ALBERTO GRÜNBAUM AND PLAMEN ILIEV

all x, y as well as all t . For a few examples of this see [15], as well as referencesin [3]. See also [24] where the author points to [21] for useful connections betweenthe heat equation and KdV invariants. For a general discussion of the bispectralproblem see [10] and [27] and for a sample of problems touching upon this areasee [4–7, 9, 11, 13].

In the case above the potentials V s are the rational solutions of the KdV equa-tion decaying at infinity, see [2]. They can be obtained by a finite number ofapplications of the Darboux process starting from the trivial potential V = 0,see [1]. The remarks above rest on the basic fact that the Darboux process mapsoperators of the form d2/dx2 + V into themselves.

Following some preliminary explorations in [14] we would like to use this‘soliton technology’ as a tool for the discovery of the general form of a heatkernel expansion on the integers. In this case, there seems to be no general theorypredicting even the existence of asymptotic expansions.

If one replaces the real line by the integers and looks for the fundamentalsolution of

ut (n, t) = u(n + 1, t) − 2u(n, t) + u(n − 1, t) ≡ L0u(n, t)

with u(n, 0) = δnm, one obtains, in terms of the Bessel function In(t) of imaginaryargument, the well-known result

u(n, t) = e−2t In−m(2t).

A nice reference for this is [12]. In the context of l2(Z) it is simplest to study L0 byFourier methods and obtain for the fundamental solution of the heat equation theexpression

e−2t

2πi

∮et (x+x−1)xn−m dx

x. (1.1)

One can replace the second difference operator L0 above by an appropriateperturbation of it and look at the corresponding heat equation and its fundamentalsolution. The purpose of this paper is to describe the result when the ordinarysecond difference operator is subject to a finite number of Darboux factorizationsteps. In this case the spectrum is a finite interval and the factorization can beperformed at each end (see (2.1)). Formula (1.1) above will be modified properly, in(5.3), when L0 is subject to a finite number of applications of the Darboux process.

We close this introduction with the simplest nontrivial example and then wedescribe the organization of the paper.

Let us write the operator L0 in the form L0 = � − 2 Id + �−1, where � standsfor the customary shift operator, acting on functions of a discrete variable n ∈ Z

by �f (n) = f (n + 1). We can factorize L0 as

L0 = P0Q0, (1.2)

HEAT KERNEL EXPANSIONS ON THE INTEGERS 185

where the operators P0 and Q0 are given by

P0 = Id − τn−1(δ)

τn(δ)�−1 and Q0 = � − τn+1(δ)

τn(δ)Id, (1.3)

with τn(δ) = n+δ. Applying one Darboux step with parameter δ to the operator L0

amounts to producing a new operator L1,0 by exchanging the order of the factorsin (1.2), i.e.

L1,0 = Q0P0 = � −(

2 + 1

τn(δ)τn+1(δ)

)Id + τn+1(δ)τn−1(δ)

τn(δ)2

�−1. (1.4)

The fundamental solution to

ut = L1,0u (1.5)

is given by

u(n,m, t) = e−2t

τm+1τn[τmτn+1In−m(2t) − tIn−m(2t) − tIn−m+1(2t)]. (1.6)

Indeed, it is obvious that u(n,m, 0) = δnm, and Equation (1.5) can be verified usingwell known identities satisfied by the Bessel functions (see (4.2a) and (4.2b)).

It is important to point out that, when the real line is replaced by the integers,the operators obtained from L0 by the Darboux process are no longer of the formL0 plus a potential.

The operators L, obtained from L0 by a finite number of applications of theDarboux process at the ends of the spectrum, have been recently determined, see[16, 17]. These operators belong to a rank-one commutative ring AV of differenceoperators with unicursal spectral curve. The common eigenfunction pn(x) to alloperators of AV , with spectral parameter x, is also an eigenfunction to a rank-onecommutative ring of differential operators in x with solitonic spectral curve, i.e. wehave a difference-differential bispectral situation. Moreover, the functions pn(x)

satisfy an orthogonality relation on the circle. These results will be summarized inthe next section and put to use in later ones.

The ring of operators AV is also discussed in Section 2 and certain operators init are exhibited in Section 3. In Section 4 we collect some properties of Bessel func-tions and then all of this is used in Section 5 to obtain our main result, Theorem 5.1.We close the paper with one example to illustrate all the steps of the proof of ourmain result what covers all cases when L belongs to a rank one bispectral ring.

2. Rank-One Bispectral Second-Order Difference Operators

Denote by � and �, respectively, the customary shift and difference operators,acting on functions of a discrete variable n ∈ Z by

�f (n) = f (n + 1) and �f (n) = f (n + 1) − f (n) = (� − Id)f (n).


The formal adjoint to an operator

X =∑j

aj (n)�j

is defined to be

X∗ =∑j

�−j · aj (n) =∑j

aj (n − j)�−j .

In this section we describe the operator LR,S obtained by successive Darbouxtransformations from the operator L0 = � − 2 Id + �−1. The symmetric operatorL0 admits (as d2/dx2 did in the case of the real line) a unique selfadjoint extensionin l2(Z) (‘limit point’ case of Weyl’s classification at both end points). Its spectrumis the interval (−4, 0). The steps of the Darboux transformations are as follows

L0 = P0Q0 � L1,0 = Q0P0 = P1Q1 � · · · � LR,0 = QR−1PR−1,

LR,0 + 4 Id = PRQR � LR,1 + 4 Id = QRPR

= PR+1QR+1 � · · · � LR,S + 4 Id = QR+S−1PR+S−1. (2.1)

At each step we perform a lower-upper factorization, as we did in (1.3), of thecorresponding operator and then we produce a new operator by interchanging thefactors. The operator LR,S is obtained by performing R Darboux steps at one endof the spectrum of L0 and S steps at the other end.

For details of the exposition that follows we refer the reader to [17].Let e(k, λ) be the linear functional acting on a function g(z) by the formula

〈e(k, λ), g〉 = g(k)(λ), λ ∈ C, k � 0.

We shall denote by Exp(n; r, z) the exponential function

Exp(n; r, z) = (1 + z)n exp

( ∞∑i=1

rizi

),

where r = (r1, r2, . . .). Let us introduce also the functions

Sεj (n; r) = 1

j ! 〈e(j, ε − 1),Exp(n; r, z)〉, (2.2)

and write

φj (n; r) = S12j−1(n + j − 1; r) and ψj(n; r) = S−1

2j−1(n + j − 1; r). (2.3)

When ε = 1, the functions S1j (n; r) are a shifted version of the classical elementary

Schur polynomials, defined by exp(∑∞

j=1 rj zj ) = ∑∞

j=0 Sj (r)zj :

S1j (n; r) = Sj (r1 + n, r2 − n/2, r3 + n/3, . . .).


For ε = −1, the functions S−1j (n; r) are of the form

S−1j (n; r) = (a polynomial in n, r1, r2, . . .) × (−1)n exp

( ∞∑j=1

(−2)j rj

).

Finally, let us define

τ(n; r) = Wr�(φ1(n; r), . . . , φR(n; r), ψ1(n; r), . . . , ψS(n; r)) ×× (−1)nS exp

(−S

∞∑i=1

ri(−2)i), (2.4)

where Wr� denotes the discrete Wronskian with respect to the variable n:

Wr�(f1(n), f2(n), . . . , fk(n)) = det(�i−1fj (n))1�i,j�k.

The purpose of the exponential factor in (2.4) is to cancel the exponential factorthat comes out from the functions ψj(n; r), 1 � j � S. With this normalization,the function τ(n; r) becomes a quasipolynomial in the variables n, r1, r2, . . ., i.e.in general, τ(n, r) depends on infinitely many variables, but there exists a positiveinteger N , such that τ(n, r) is a polynomial in every variable of degree at most N .The operator LR,S in (2.1) can be expressed in terms of the function τ(n, r) via theformula

LR,S = � +(

−2 + ∂

∂r1log

τ(n + 1; r)τ(n; r)

)Id +

+ τ(n − 1; r)τ(n + 1; r)τ(n; r)2

�−1. (2.5)

From (2.4) it follows immediately that τ(n, r) is an adelic tau function+ of the�KP hierarchy defined by R + S one-point conditions, with R conditions at thepoint zero and S conditions at the point −2. The reason why the end points of thespectrum of L0, 0 and −4, are replaced by 0 and −2 can be traced back to (2.2)and (2.3). After a suitable linear change of time variables r1, r2, . . . the tridiagonaloperator LR,S solves the Toda lattice hierarchy, i.e.

∂L

∂r ′j

= [(Lj )+, L], j = 1, 2, . . . ,

where (Lj )+ denotes the positive difference part++ of the operator Lj , and {r ′j }∞

j=1are related to {rj }∞

j=1 via linear transformation of the form

rj = r ′j +

∞∑k=j+1

cjkr′k,

+ By an adelic tau function we mean a tau function built from a plane belonging to Wilson’s adelicGrassmannian [27] via the construction in [16].++ Equivalently, if we think of Lj as an infinite matrix, (Lj )+ is the upper part of this matrix,

including the main diagonal.


see [16] for explicit formulas. The wave function w(n; r, z) and the adjoint wavefunction w∗(n; r, z) are defined by

w(n; t, z) = τ(n; r − [z−1])τ (n; r) Exp(n; r, z)

=(

1 +∞∑j=1

wj(n; r)z−j

)Exp(n; r, z) (2.6)

and

w∗(n; r, z) = τ(n; r + [z−1])τ (n; r) Exp−1(n; r, z)

=(

1 +∞∑j=1

w∗j (n; r)z−j

)Exp−1(n; r, z), (2.7)

where [z] = (z, z2/2, z3/3, . . .). The wave operator W(n; r) and the adjoint waveoperator (W−1)∗ can be written in the form

W(n; r) = 1 +∞∑j=1

wj(n; r)�−j = Q(� − Id)−R(� + Id)−S, (2.8)

(W−1)∗(n; r) = 1 +∞∑j=1

w∗j (n + 1; r)�∗ −j

= P ∗(�∗ − Id)−R(�∗ + Id)−S, (2.9)

where P and Q are finite-band forward difference operators of order R + S satis-fying

PQ = (� − Id)2R(� + Id)2S, (2.10)

see [17, Theorem 3.3].The tau function τ(n; r) can be viewed as an infinite sequence (indexed by

n) of tau functions of the standard KP hierarchy, associated with a flag of nestedsubspaces

V: · · · ⊂ Vn+1 ⊂ Vn ⊂ Vn−1 ⊂ · · · , (2.11)

with

Vn = Span{w(n; 0, z), w(n + 1; 0, z), w(n + 2; 0, z), . . .}. (2.12)

The plane V0 belongs to Wilson’s adelic Grassmannian [27]. Let x = z + 1,

pn(x) = W(n; r)xn = w(n, r, x − 1) exp

(−

∞∑j=1

rj (x − 1)j), (2.13)


and similarly

p∗n(x) = W ∗(n − 1; r)x−n = w∗(n, r, x − 1) exp

( ∞∑j=1

rj (x − 1)j). (2.14)

From (2.1) it follows easily that pn(x) are eigenfunctions of the operator LR,S witheigenvalue x − 2 + x−1, i.e.

LR,Spn(x) = (x − 2 + x−1)pn(x). (2.15)

Moreover, pn(x) are also eigenfunctions of a differential operator in x (with co-efficients independent of n) and thus provide a difference-differential analog ofthe rank-one solutions (the KdV family) of the bispectral problem considered byDuistermaat and Grünbaum in [10].

Below we discuss the spectral curve and the common eigenfunction of themaximal commutative ring of difference operators AV containing LR,S . The cor-respondence between commutative rings of difference operators and curves wasstudied by van Moerbeke and Mumford [26] and Krichever [19] (see also [22]where the case of singular curves was treated very completely).

Let us introduce the ring AV of Laurent polynomials in x that preserve the flagV:

AV = {f (x) ∈ C[x, x−1] : f (x)Vn ⊂ Vn+k,

for some k ∈ Z and ∀n ∈ Z}. (2.16)

For each f (x) ∈ AV there exists a finite band operator Lf , such that

Lfpn(x) = f (x)pn(x).

Moreover, if f (x) = ∑m2j=m1

ajxj with am1 �= 0 and am2 �= 0, then the operator Lf

has support [m1,m2], i.e.

Lf =m2∑

j=m1

bj (n)�j .

We can think of Lf as an operator obtained by a Darboux transformation from theconstant coefficient operator f (�). The ring AV is generated by the functions+

w = 1

2

(x + 1

x

)and v = 1

2R+S+1

(x − 1)2R+1(x + 1)2S+1

xR+S+1, (2.17)

i.e. AV = C[w, v]. Thus the operator LR,S constructed above belongs to a maximalrank-one commutative ring of difference operators AV isomorphic to the ring ofLaurent polynomials AV . The spectral curve of AV is

Spec(AV): v2 = (w − 1)2R+1(w + 1)2S+1.

+ Notice that these generators differ slightly from the ones chosen in [17].


The operators Q and P ∗ in formulas (2.8) and (2.9), can be expressed in termsof the functions {φi(n; r)}Ri=1 and {ψj(n; r)}Sj=1 as

Qf (n) = Wr�(φ1(n; r), . . . , φR(n; r), ψ1(n; r), . . . , ψS(n; r), f (n))

Wr�(φ1(n; r), . . . , φR(n; r), ψ1(n; r), . . . , ψS(n; r)) (2.18)

and

P ∗ f (n) = Wr�∗(φ∗1 (n; r), . . . , φ∗

R(n; r), ψ∗1 (n; r), . . . , ψ∗

S (n; r), f (n))

Wr�∗(φ∗1 (n; r), . . . , φ∗

R(n; r), ψ∗1 (n; r), . . . , ψ∗

S (n; r)) , (2.19)

with

φ∗i (n; t) = φi(n + R + S; t), 1 � i � R,

and

ψ∗j (n; t) = ψj(n + R + S; t), 1 � j � S.

Using these explicit formulas for Q and P , one can check that

pn(x−1) = τ(n + 1; r)

τ(n; r) xp∗n+1(x). (2.20)

Finally, (see [17, Theorem 5.2]) one can prove the following orthogonality relation:

1

2πi

∮C

pn(x)pm(x−1)

dx

x= τ(n + 1; t)

τ (n; t) δnm, ∀n,m ∈ Z, (2.21)

where C is any positively oriented simple closed contour surrounding the origin,not passing through the points x = ±1. Indeed, pn(x) are rational functions on theRiemann sphere with poles only at x = 0,±1,∞. The spectral curve Spec(AV)

has cusps at ±1, hence using the fact that the residue of a regular differential at acusp is always zero (see [25]), we deduce that

resx=±1 pn(x)p∗m(x) dx = 0.

The proof of (2.21) now follows from the discrete Kadomtsev–Petviashvili bilinearidentities and the relation between the wave and the adjoint wave functions in(2.20).

3. Certain Laurent Polynomials in AV

The main result in this section is Proposition 3.1 below which allows us to constructsome ‘universal’ Laurent polynomials in AV . This result is crucial for the proof ofTheorem 5.1. As we shall see in Section 5, applying Proposition 3.1, we can reducethe infinite sum in the computation of the fundamental solution of (5.1)–(5.2) to afinite sum, modulo some identities among the Bessel functions to be discussed inthe next section.


PROPOSITION 3.1. Let T be � max(R, S), and let s0, s1, . . . , sT be distinctnonzero integers, such that sj ≡ sk (mod 2) and sj + sk �= 0, for 0 � j, k � T .Then

Fs0,...,sT (x) =T∑

k=0

xsk

sk∏

j �=k(s2k − s2

j )∈ AV . (3.1)

The proof of Proposition 3.1 is based on two simple lemmas related to theLagrange interpolation polynomial and the Chebyshev polynomials of the secondkind.

LEMMA 3.2. Let q(n) be an odd polynomial in n of degree at most 2T − 1, andlet s0, s1, . . . , sT be distinct positive numbers. Then

T∑k=0

q(sk)

sk∏

j �=k(s2k − s2

j )= 0. (3.2)

Proof of Lemma 3.2. Consider the Lagrange interpolation polynomial for q(n) atthe nodes ±s0,±s1, . . . ,±sT . Since deg q(n) � 2T − 1, the Lagrange polynomialmust be identically equal to q(n), i.e. we have

T∑k=0

q(sk)n + sk

2sk

∏j �=k

n2 − s2j

s2k − s2

j

+T∑

k=0

q(−sk)n − sk

−2sk

∏j �=k

n2 − s2j

s2k − s2

j

= q(n).

Hence, the coefficient of n2T+1 on left-hand side must be zero, which gives (3.2). ✷The Chebyshev polynomials of the second kind are defined by the following

three term recurrence relation

2wUn(w) = Un+1(w) + Un−1(w), n = 1, 2, . . . , (3.3)

with U0(w) = 1 and U1(w) = 2w. From the last formula one sees immediately thatUn(w) is an even/odd polynomial if n is an even/odd positive integer, respectively.

The Chebyshev polynomials of the second kind can be obtained from the Jacobipolynomials by taking α = β = 1/2:

Un(w) = (n + 1)2F1

(−n, n + 232

∣∣∣ 1 − w

2

)

= (n + 1)n∑

k=0

2k

(2k + 1)!

(k∏

j=1

((n + 1)2 − j 2)

)(w − 1)k. (3.4)

These explicit expressions for Un(w) will be needed below.


LEMMA 3.3. If s0, s1, . . . , sT are distinct positive integers, then the polynomial

Qs0,...,sT (w) =T∑

k=0

Usk−1(w)

sk∏

j �=k(s2k − s2

j )(3.5)

is divisible by (w − 1)T . Moreover if s0 ≡ s1 ≡ · · · ≡ sT (mod 2), then

(w2 − 1)T /Qs0,...,sT (w). (3.6)

Proof. From (3.4) we see that

U(k)n−1(1) = 2kk!

(2k + 1)!nk∏

j=1

(n2 − j 2),

i.e. U(k)n−1(1) is an odd polynomial in n of degree 2k+1. From Lemma 3.2 it follows

that

Q(k)s0,...,sT

(1) = 0, k = 0, 1, . . . , T − 1,

hence (w − 1)T /Qs0,...,sT (w). If s0 ≡ s1 ≡ · · · ≡ sT (mod 2), then Qs0,...,sT (w) iseither even or odd polynomial, which proves (3.6). ✷

Proof of Proposition 3.1. Notice that

Fs0,...,sk−1,sk,sk+1,...,sT (x) − Fs0,...,sk−1,−sk,sk+1,...,sT (x)

= 1

sk∏

j �=k(s2k − s2

j )(xsk + x−sk ) ∈ AV .

Thus, without any restriction, we may assume that s0, . . . , sT are positive integers.We can rewrite the first equation in (2.17) as x2 = 2wx − 1. From this relation,one can easily see by induction that

xk = Uk−1(w)x − Uk−2(w), for k = 1, 2, . . . , (3.7)

with the understanding that U−1(w) = 0. Using (3.7), we can write the function in(3.1) in the form

Fs0,...,sT (x) = Qs0,...,sT (w)x +T∑

k=0

Usk−2(w)

sk∏

j �=k(s2k − s2

j ), (3.8)

where Qs0,...,sT (w) is the polynomial defined by (3.5). Clearly, the sum in (3.8)belongs to AV , so it remains to show that

Qs0,...,sT (w)x ∈ AV . (3.9)


Using (2.17) we can write v in terms of w and x as

v = (w − 1)R(w + 1)S(x − w),

which gives

(w − 1)R(w + 1)Sx = v + w(w − 1)R(w + 1)S ∈ AV .

Therefore, if T � max(R, S), we have

(w2 − 1)T x ∈ AV . (3.10)

The proof now follows from Lemma 3.3. ✷

4. Some Identities Satisfied by the Bessel Functions

The Bessel functions of imaginary argument are defined by the generating function∑k∈Z

Ik(t)xk = et (x+x−1)/2, (4.1)

whence Ik(t) = I−k(t), and Ik(−t) = (−1)kIk(t). Differentiating (4.1) with re-spect to x and t , one gets

kIk(t) = t

2(Ik−1(t) − Ik+1(t)) (4.2a)

and

I ′k(t) = 1

2 (Ik−1(t) + Ik+1(t)) , (4.2b)

respectively. Similarly, one can show that Ik(t) satisfy the modified Bessel equation

(t2∂2t + t∂t − (t2 + k2))Ik(t) = 0, (4.3)

where ∂t = d/dt . The main result in this section is Proposition 4.1 below, whichsays that if q(j) is an odd polynomial of j , then the sum∑

j>kj odd/even

q(j)Ij (t),

can be written as a finite linear combination of Bessel functions of the form∑finitely many j’s

αj(t)Ij (t),

where each coefficient αj(t) is a polynomial in t . Let us define a sequence ofpolynomials αn

j (t) for j ∈ Z and n = 0, 1, . . . as follows:

α0j (t) = δj0

t

2={t/2 if j = 0,0 if j �= 0

(4.4)


and

αn+1j (t) = (t2∂2

t + t∂t )αnj (t) +

(t2∂t + t

2

)(αn

j+1(t) + αnj−1(t)) +

+ t2

4(αn

j+2(t) − 2αnj (t) + αn

j−2(t)). (4.5)

From the symmetry of the defining relation, it follows that αnj (t) = αn

−j (t). More-over, it is clear that

αnj = 0 for j /∈ [−2n, 2n], (4.6)

and αn−2n(t) = αn

2n(t) = t2n+1/22n+1. For arbitrary j ∈ [−2n, 2n], αnj (t) is a

polynomial in t of degree at most 2n + 1.

PROPOSITION 4.1. Let k and n be integers, n � 0. Then

∑j>k

j≡k+1 (mod 2)

j 2n+1Ij (t) =∑s∈Z

αns−k(t)Is(t) =

2n+k∑s=−2n+k

αns−k(t)Is(t). (4.7)

Proof by induction. For n = 0 we have to show that∑j>k

j≡k+1 (mod 2)

jIj (t) = t

2Ik(t), (4.8)

which easily follows from (4.2a). Assume that (4.7) holds for some n. Then, usingthe modified Bessel equation (4.3) we obtain

∞∑l=0

(k + 2l + 1)2n+3Ik+2l+1(t)

= (t2∂2t + t∂t − t2)

∞∑l=0

(k + 2l + 1)2n+1Ik+2l+1(t)

= (t2∂2t + t∂t − t2)

∑s∈Z

αns−k(t)Is(t),

which shows that (4.7) holds for n + 1, upon using (4.2b). ✷

5. Heat Kernel Expansions on the Integers

In the case of the real line, the solution of the heat equation is not unique unlessthe class of solutions satisfies a condition of the form

|u(x, t)| � c1ec2x2, c1, c2 � 0.


When one says that the Gaussian kernel is the fundamental solution of the heatequation one (implicitly) assumes that one is considering functions of moderategrowth at infinity. Denote by u(n,m, t) the solution of

ut = LR,Su, (5.1)

u|t=0 = δnm, (5.2)

where LR,S is the second-order difference operator (acting on functions of a dis-crete variable n ∈ Z) constructed in Section 2. We make the same implicit assump-tion here when we say that u(n,m, t) is the fundamental solution.

From (2.15) and (2.21) and elementary spectral theory, it follows that

u(n,m, t) = τ(m)

τ(m + 1)

e−2t

2πi

∮C

et (x+x−1)pn(x)pm(x−1)

dx

x, (5.3)

where, for simplicity, we have omitted the dependence on the parameters r =(r1, r2, . . .). Using (2.20) we can write also the fundamental solution in terms ofthe wave and the adjoint wave functions as

u(n,m, t) = e−2t

2πi

∮C

et (x+x−1)pn(x)p∗m+1(x) dx. (5.4)

THEOREM 5.1. The solution of (5.1) with initial condition (5.2) can be writtenin the form

u(n,m, t) = e−2t∑

finitely many j’s

βj(n,m, t)Ij (2t), (5.5)

where βj(n,m, t) are polynomials in t of degree at most 2T − 1, with T =max(R, S).

Proof. From (2.20) it follows that

u(n,m, t) = τ(m)τ(n + 1)

τ (m + 1)τ (n)u(m, n, t).

Thus, we can assume that k = n − m � 0. Around x = 0, we have the expansion

pn(x)pm(x−1)

x= τ(m + 1)

τ (m)pn(x)p

∗m+1(x)

= xn−m−1

τ(n)τ(m)(τ(n + 1)τ (m) + O(x)),

which shows that

resx=0

(xjpn(x)pm(x

−1)dx

x

)= 0 for j � 1 − k. (5.6)


For ε = 1, 2 and i = 0, 1, . . . , T − 1 denote

qk+ε+2i (j) = j

k + ε + 2i

T∏l=1l �=i

j 2 − (k + ε + 2l)2

(k + ε + 2i)2 − (k + ε + 2l)2. (5.7)

qk+ε+2i (j) is an odd polynomial in j of degree 2T − 1. We have

et (x+x−1) =∑

j�1−k

Ij (2t)xj + Ik(2t)x

−k + (5.8a)

+∑

j>k+2T

Ij (2t)

(x−j −

T−1∑i=0

qk+ε+2i (j)x−(k+ε+2i)

)+ (5.8b)

+2∑

ε=1

T−1∑i=0

[ ∑j�k+ε+2i

j≡k+ε (mod 2)

qk+ε+2i (j)Ij (2t)

]x−(k+ε+2i), (5.8c)

where in the second sum, for every fixed j > k + 2T , we choose ε = 1 or 2, sothat j ≡ k+ε (mod 2). Denote by fj (x) the Laurent polynomial of x in the secondsum in (5.8), i.e.

fj (x) = x−j −T−1∑i=0

qk+ε+2i(j)x−(k+ε+2i).

Notice that

fj (x) = −j

T∏l=1

(j 2 − (k + ε + 2l)2)Fs0,s1,...,sT (x),

where Fs0,s1,...,sT (x) is the function defined by (3.1) with si = −(k + ε + 2i),i = 0, 1, . . . , T − 1 and sT = −j . Thus, by Proposition 3.1, fj (x) ∈ AV ,and therefore there exists a difference operator Lfj with support [−j,−(k + ε)],satisfying fj (x)pn(x) = Lfjpn(x). Since n − (k + ε) = m − ε < m we see that∮

C

fj (x)pn(x)pm(x−1)

dx

x= 0. (5.9)

From (5.3), (5.6) and (5.9) it follows that the infinite sums in (5.8a) and (5.8b)do not ‘contribute’ to the fundamental solution u(n,m, t). Finally, the infinitesum in (5.8c) can be rewritten as a finite linear combination of Bessel functionswith polynomial coefficients, according to Proposition 4.1, which completes theproof. ✷

We shall illustrate all steps of the proof by considering the case R = S = 1.


EXAMPLE 5.2. Let R = S = 1. From (2.2) and (2.3) one computes

φ1(n; r) = S11(n; r) = n + r1;

ψ1(n; r) = S−11 (n; r) =

(−n +

∞∑j=1

(−2)j−1jrj

)(−1)n exp

( ∞∑j=1

(−2)j rj

).

Denote for simplicity α = r1 and β = ∑∞j=2(−2)j−1jrj . The tau function is given

by formula (2.4):

τ(n) =∣∣∣∣ n + α −n + α + β

1 2n + 1 − 2(α + β)

∣∣∣∣ . (5.10)

The second-order difference operator L1,1 is given by formula (2.5), with ∂/∂r1 =∂/∂α.

From (2.6), (2.8), (2.13) and (2.18) we get

pn(x) = xn

τ(n)(x2 − 1)

∣∣∣∣∣∣n + α −n + α + β 1

n + 1 + α n + 1 − α − β x

n + 2 + α −n − 2 + α + β x2

∣∣∣∣∣∣ . (5.11)

From the last formula one can easily deduce that near x = 0 we can expandpn(x)pm(x

−1) as

xn−m

(τ(n + 1)

τ (n)+

∞∑j=1

γjxj

), (5.12)

where

γj = − 4

τ(n)τ(m)[(m − α − β + 1)(n − α − β + 1)(n − m + j) ++ (−1)j (m + α + 1)(n + α + 1)(n − m + j)]. (5.13)

From (5.13) and (4.8) one can see that indeed u(n,m, t) is a finite linear combina-tion of Bessel functions. Below, we shall illustrate how this can be seen followingthe proof of Theorem 5.1 using just the first few coefficients in (5.12).

Proposition 3.1 tells us that ∀s, l �= 0, such that s ≡ l (mod 2)

xs

s− xl

l∈ AV .

Following (5.8), we can write et (x+x−1) as

et (x+x−1) =∑

j�1−k

Ij (2t)xj + Ik(2t)x

−k + (5.14a)

+∑j>k+2

j≡k+1 (mod 2)

Ij (2t)

(x−j − j

k + 1x−(k+1)

)+ (5.14b)


+∑j>k+2

j≡k (mod 2)

Ij (2t)

(x−j − j

k + 2x−(k+2)

)+ (5.14c)

+ 1

k + 1

[ ∑j>k

j≡k+1 (mod 2)

jIj (2t)

]x−(k+1) + (5.14d)

+ 1

k + 2

[ ∑j>k+1

j≡k (mod 2)

jIj (2t)

]x−(k+2), (5.14e)

where k = n − m. Using (4.8), we can write the sums in (5.14d) and (5.14e) as∑j>k

j≡k+1 (mod 2)

jIj (2t) = tIk(2t) and∑j>k+1

j≡k (mod 2)

jIj (2t) = tIk+1(2t).

Thus (5.14) can be rewritten as

et (x+x−1) =∑

j�1−k

Ij (2t)xj + (5.15a)

+∑j>k+2

j≡k+1 (mod 2)

Ij (2t)

(x−j − j

k + 1x−(k+1)

)+ (5.15b)

+∑j>k+2

j≡k (mod 2)

Ij (2t)

(x−j − j

k + 2x−(k+2)

)+ (5.15c)

+ x−k

[Ik(2t) + t

k + 1Ik(2t)x

−1 + t

k + 2Ik+1(2t)x

−2

]. (5.15d)

The sums in (5.15a), (5.15b) and (5.15c) ‘do not contribute’ to the integral (5.3)(see the proof of Theorem 5.1). Thus

u(n,m, t)

= e−2t resx=0

[(Ik(2t) + t

k + 1Ik(2t)x

−1 + t

k + 2Ik+1(2t)x

−2

)×

× τ(m)

τ(m + 1)

pn(x)pm(x−1)

xk+1

]

= e−2t τ (m)

τ(m + 1)

(τ(n + 1)

τ (n)Ik(2t) + γ1

k + 1tIk(2t) + γ2

k + 2tIk+1(2t)

), (5.16)

where γ1 and γ2 are the coefficients in the expansion (5.12). Using (5.13) we getthe following explicit formula for u(n,m, t)

u(n,m, t) = e−2t

τ (m + 1)τ (n)[τ(m)τ(n + 1)In−m(2t) +


+ 4t (β + 2α)(n + m − β + 2)In−m(2t) −− 4t (2mn − βn + 2n − βm + 2m + β2 ++ 2αβ − 2β + 2α2 + 2)In−m+1(2t)]. (5.17)

If we put δ = α + 1/2 and let β → ∞ we get

u(n,m, t) = e−2t

τm+1τn[τmτn+1In−m(2t) − tIn−m(2t) − tIn−m+1(2t)],

where τn = n + δ. Notice that this is exactly formula (1.6) for the fundamentalsolution computed in the introduction (R = 1, S = 0).

The referee has raised the interesting possibility of using the formula

etAB = 1 + A

(∫ t

0esBA ds

)B

to get an alternative proof of Theorem 5.1. This remains a challenging problem.

Acknowledgements

We thank Henry P. McKean and the referee for suggestions that led to an improvedversion of this paper.

References

1. Adler, M. and Moser, J.: On a class of polynomials connected with the Korteweg–de Vriesequation, Comm. Math. Phys. 61 (1978), 1–30.

2. Airault, H., McKean, H. P. and Moser, J.: Rational and elliptic solutions of the Korteweg–de Vries equation and a related many-body problem, Comm. Pure Appl. Math. 30 (1977), 95–148.

3. Avramidi, I. G. and Schimming, R.: Heat kernel coefficients for the matrix Schrödingeroperator, J. Math. Phys. 36 (1995), 5042–5054.

4. Berest, Y.: Huygens principle and the bispectral problem, In: The Bispectral Problem(Montreal, PQ, 1997), CRM Proc. Lecture Notes 14, Amer. Math. Soc., Providence, RI, 1998,pp. 11–30.

5. Berest, Y. and Kasman, A.: D-modules and Darboux transformations, Lett. Math. Phys. 43(1998), 279–294.

6. Berest, Y. and Veselov, A. P.: The Huygens principle and integrability, Uspekhi Mat. Nauk 49(1994), 7–78, transl. in Russian Math. Surveys 49 (1994), 5–77.

7. Berest, Y. and Wilson, G.: Classification of rings of differential operators on affine curves,Internat. Math. Res. Notices 2 (1999), 105–109.

8. Berline, N., Getzler, E. and Vergne, M.: Heat Kernels and Dirac Operators, Grundlehren Math.Wiss. 298, Springer-Verlag, Berlin, 1992.

9. Chalykh, O. A., Feigin, M. V. and Veselov, A. P.: Multidimensional Baker–Akhiezer functionsand Huygens’ principle, Comm. Math. Phys. 206 (1999), 533–566.

10. Duistermaat, J. J. and Grünbaum, F. A.: Differential equations in the spectral parameter, Comm.Math. Phys. 103 (1986), 177–240.


11. Felder, G., Markov, Y., Tarasov, V. and Varchenko, A.: Differential equations compatible withKZ equations, Math. Phys. Anal. Geom. 3 (2000), 139–177.

12. Feller, W.: An Introduction to Probability Theory and Its Applications, Vol. 2, Wiley, New York,1990.

13. Granovskii, Ya. I., Lutzenko, I. M. and Zhedanov, A. S.: Mutual integrability, quadraticalgebras, and dynamical symmetry, Ann. Phys. 217 (1992), 1–20.

14. Grünbaum, F. A.: The bispectral problem: an overview, In: J. Bustoz et al. (eds), SpecialFunctions 2000: Current Perspective and Future Directions, 2001, pp. 129–140.

15. Grünbaum, F. A.: Some bispectral musings, In: The Bispectral Problem (Montreal, PQ, 1997),CRM Proc. Lecture Notes 14, Amer. Math. Soc., Providence, RI, 1998, pp. 11–30.

16. Haine, L. and Iliev, P.: Commutative rings of difference operators and an adelic flag manifold,Internat. Math. Res. Notices 6 (2000), 281–323.

17. Haine, L. and Iliev, P.: A rational analogue of the Krall polynomials, In: Kowalevski Workshopon Mathematical Methods of Regular Dynamics (Leeds, 2000), J. Phys. A: Math. Gen. 34(2001), 2445–2457.

18. Kac, M.: Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966), 1–23.19. Krichever, I. M.: Algebraic curves and non-linear difference equations, Uspekhi Mat. Nauk 33

(1978), 215–216, transl. in Russian Math. Surveys 33 (1978), 255–256.20. McKean, H. P. and Singer, I.: Curvature and the eigenvalues of the Laplacian, J. Differential

Geom. 1 (1967), 43–69.21. McKean, H. P. and van Moerbeke, P.: The spectrum of Hill’s equation, Invent. Math. 30 (1975),

217–274.22. Mumford, D.: An algebro-geometric construction of commuting operators and of solutions

to the Toda lattice equation, Korteweg–de Vries equation and related non-linear equa-tions, In: M. Nagata (ed.), Proceedings of International Symposium on Algebraic Geometry(Kyoto 1977), Kinokuniya Book Store, Tokyo, 1978, pp. 115–153.

23. Rosenberg, S.: The Laplacian on a Riemannian Manifold. An Introduction to Analysis onManifolds, London Math. Soc. Stud. Texts 31, Cambridge Univ. Press, Cambridge, 1997.

24. Schimming, R.: An explicit expression for the Korteweg–de Vries hierarchy, Z. Anal. Anwen-dungen 7 (1988), 203–214.

25. Serre, J.-P.: Groupes algébriques et corps de classes, Hermann, Paris, 1959.26. van Moerbeke, P. and Mumford, D.: The spectrum of difference operators and algebraic curves,

Acta Math. 143 (1979), 93–154.27. Wilson, G.: Bispectral commutative ordinary differential operators, J. Reine Angew. Math. 442

(1993), 177–204.


201

Classification of Gauge Orbit Types forSU(n)-Gauge Theories

G. RUDOLPH1, M. SCHMIDT1 and I. P. VOLOBUEV2

1Institute for Theoretical Physics, University of Leipzig, Augustusplatz 10, 04109 Leipzig, Germany.e-mail: [email protected], [email protected] Physics Institute, Moscow State University, 119899 Moscow, Russia

(Received: 15 February 2001; in final form: 4 March 2002)

Abstract. A method for determining the orbit types of the action of the group of gauge trans-formations on the space of connections for gauge theories with gauge group SU(n) in spacetimedimension d � 4 is presented. The method is based on the one-to-one correspondence between orbittypes and holonomy-induced reductions of the underlying principal SU(n)-bundle. It is shown thatthe orbit types are labelled by certain cohomology elements of spacetime satisfying two relations.Thus, for every principal SU(n)-bundle the corresponding stratification of the gauge orbit space canbe explicitly determined. As an application, a criterion characterizing kinematical nodes for physicalstates in Yang–Mills theory with the Chern–Simons term proposed by Asorey et al. is discussed.

Mathematics Subject Classifications (2000): 53C05, 53C80.

Key words: classification, gauge orbit space, nongeneric strata, orbit types, quantum nodes, stratifi-cation.

1. Introduction

One of the basic principles of modern theoretical physics is the principle of localgauge invariance. Its application to the theory of particle interactions gave riseto the standard model, which proved to be a success from both the theoreticaland phenomenological points of view. The most impressive results of the modelwere obtained within the perturbation theory, which works well for high energyprocesses. On the other hand, the low energy hadron physics, in particular, thequark confinement, turns out to be dominated by nonperturbative effects, for whichthere is no rigorous theoretical explanation yet.

The application of geometrical methods to non-Abelian gauge theories revealedtheir rich geometrical and topological properties. In particular, it showed that theconfiguration space of such theories, which is the space of gauge group orbits inthe space of connections, may have a highly nontrivial structure. In general, theorbit space possesses not only orbits of the so-called principal type, but also orbitsof other types, which may give rise to singularities of the configuration space. Thisstratified structure of the gauge orbit space is believed to be of importance for

202 G. RUDOLPH ET AL.

both the classical and quantum properties of non-Abelian gauge theories in thenonperturbative approach, and it has been intensively studied in recent years. Letus discuss some aspects indicating its physical relevance.

First, studying the geometry and topology of the generic (principal) stratum,one gets a deeper understanding of the Gribov ambiguity and of anomalies interms of index theorems. In particular, one gets anomalies of the purely topologicaltype, which cannot be seen by perturbative quantum field theory. These are wellknown results from the eighties. Moreover, there are partial results and conjec-tures concerning the relevance of nongeneric strata. First of all, nongeneric gaugeorbits affect the classical motion on the orbit space due to boundary conditionsand, in this way, may produce nontrivial contributions to the path integral. Theymay also lead to localization of certain quantum states, as it was suggested byfinite-dimensional examples [10]. Further, the gauge field configurations belong-ing to nongeneric orbits can possess a magnetic charge, i.e., they can be consid-ered as a kind of magnetic monopole configurations, which are responsible forquark confinement. This picture was found in three-dimensional gauge systems [3],and it is conjectured that it can hold for four-dimensional theories as well [4].Finally, it was suggested in [16] that nongeneric strata may lead to additionalanomalies.

Most of the problems mentioned here are still awaiting a systematic investiga-tion. In a series of papers, we are going to make a new step in this direction. Wegive a complete solution to the problem of determining the strata that are present inthe gauge orbit space for SU(n) gauge theories in compact Euclidean spacetime ofdimension d = 2, 3, 4. Our analysis is based on the results of a paper by Kondrackiand Rogulski [23], where it was shown that the gauge orbit space is a stratifiedtopological space in the ordinary sense (cf. [22] and references therein). Moreover,these authors found an interesting relation between orbit types and certain bundlereductions, which we are going to use. We also refer to [14] for the discussion ofa very simple, but instructive special example (orbit types of SU(2)-gauge theoryon S4).

The paper is organized as follows. In Section 2 we introduce the basic notionsrelated to the action of the group of gauge transformations on the space of con-nections, state the definitions of stabilizer and orbit type and recall basic resultsconcerning the stratification structure of the gauge orbit space. In Section 3 weintroduce holonomy-induced bundle reductions and establish their connection withorbit types. As a tool for determining such bundle reductions, we introduce thenotions of a Howe subgroup and a Howe subbundle. Section 4 is devoted to thestudy of the Howe subgroups of SU(n). In Section 5 we give a classification of theHowe subbundles of SU(n)-bundles for spacetime dimension d � 4. In Section 6we prove that any Howe subbundle of SU(n)-bundles is holonomy-induced. In Sec-tion 7 we implement the equivalence relation of Howe subbundles due to the actionof SU(n). As an example, in Section 8 we determine the orbit types for gauge groupSU(2). Finally, in Section 9 we discuss an application to Chern–Simons theory in

CLASSIFICATION OF GAUGE ORBIT TYPES 203

2+1 dimensions. In two subsequent papers, we shall investigate the natural partialordering on the set of orbit types and the structure of another, coarser stratification(see [24]) obtained by first factorizing with respect to the so-called pointed gaugegroup and then by the structure group.

2. Gauge Orbit Types and Stratification

We consider a fixed topological sector of a gauge theory with gauge group G ona Riemannian manifold M. Within the geometrical setting, it means that we aregiven a smooth right principal fibre bundle P with structure group G over M. G isassumed to be a compact connected Lie group and M is assumed to be compact,connected, and orientable.

Denote the sets of connection forms and gauge transformations of P of Sobolevclass Wk by Ak and Gk, respectively. For generalities on Sobolev spaces of cross-sections in fibre bundles, see [28]. Provided 2k > dim M, Ak is an affine Hilbertspace and Gk+1 is a Hilbert Lie group acting smoothly from the right on Ak [23,26, 32]. We shall even assume that 2k > dim M+2. Then, by the Sobolev Embed-ding Theorem, connection forms are of class C1 and, therefore, have continuouscurvature. If we view elements of Gk+1 as G-space morphisms P → G, the actionof g ∈ Gk+1 on A ∈ Ak is given by

A(g) = g−1Ag + g−1 dg. (1)

Let Mk denote the quotient topological space Ak/Gk+1. This space represents theconfiguration space of our gauge theory.

For this to make sense, Mk should not depend essentially on the purely tech-nical parameter k. Indeed, let k′ > k. Then one has natural embeddings Gk′+1 ↪→Gk+1 and Ak′ ↪→ Ak. As a consequence of the first, the latter projects to a mapϕ: Mk′ →Mk. Since the image of Ak′ in Ak is dense, so is ϕ(Mk′) in Mk. To seethat ϕ is injective, let A1, A2 ∈ Ak′ and g ∈ Gk+1 such that A2 = A

(g)

1 . Then (1)implies

dg = gA2 − A1g. (2)

Due to 2k′ > 2k > dim M, by the multiplication rule for Sobolev functions, theright-hand side of (2) is of class Wk+1. Then g is of class Wk+2. This can be iterateduntil the right-hand side is of class Wk′ . Hence, g ∈ Gk′+1, so that A1 and A2 arerepresentatives of the same element of Mk′ . This shows that Mk′ can be identifiedwith a dense subset of Mk. Another question is whether the stratification structureof Mk, which will be discussed in a moment, depends on k. Fortunately, the answerto this question is negative, see Theorem 3.3.

In general, the orbit space of a smooth Lie group action does not admit a smoothmanifold structure. The best one can expect is that it admits a stratification. For thenotion of stratification of a topological space, see [22] or [23, § 4.4]. For the gaugeorbit space Mk, a stratification was constructed in [23], using a method which is


known from compact Lie group actions on completely regular spaces [7]. In orderto explain this, let us recall the notions of stabilizer and orbit type. The stabilizer,or isotropy subgroup, of A ∈ Ak is the subgroup

Gk+1A = {g ∈ Gk+1 | A(g) = A}

of Gk+1. It has the following transformation property: For any A ∈ Ak and g ∈Gk+1,

Gk+1A(g) = g−1Gk+1

A g.

Thus, there exists a natural map, called a type map, assigning to each element of Mk

the conjugacy class in Gk+1 made up by the stabilizers of its representatives in Ak.Let �k denote the image of this map. The elements of �k are called orbit types. Theset �k carries a natural partial ordering: τ � τ ′ iff there are representatives Gk+1

A

of τ and Gk+1A′ of τ ′ such that Gk+1

A ⊇ Gk+1A′ . Note that this definition is consistent

with [7] but not with [23] and several other authors who define it just inversely.As was shown in [23], the subsets Mk

τ ⊆ Mk, consisting of gauge orbits oftype τ , can be equipped with a smooth Hilbert manifold structure and the family{Mk

τ | τ ∈ �k} is a stratification of Mk. Accordingly, the manifolds Mkτ are called

strata. In particular,

Mk =⋃

τ∈�k

Mkτ ,

where for any τ ∈ �k, Mkτ is open and dense in

⋃τ ′�τ Mk

τ ′ . Similarly to the caseof compact Lie groups, there exists a maximal orbit type τ0, called the principalorbit type. Since the corresponding stratum Mk

τ0is open and dense in Mk, τ0 and

Mkτ0

are also called generic orbit type and generic stratum, respectively.The above considerations show that the set �k, together with its natural partial

ordering, carries the information about which strata occur and how they are patchedtogether.

To conclude, let us remark that instead of using Sobolev techniques, one canalso stick to smooth connection forms and gauge transformations. Then one obtainsessentially analogous results about the stratification of the corresponding gaugeorbit space where, roughly speaking, one has to replace ‘Hilbert manifold’ and‘Hilbert Lie group’ by ‘tame Fréchet manifold’ and ‘tame Fréchet Lie group’,see [1, 2].

3. Correspondence between Orbit Types and Bundle Reductions

In this section, let p0 ∈ P be fixed. For A ∈ Ak, let HA and PA denote theholonomy group and holonomy subbundle, respectively, of A based at p0. Weassume 2k > dim M + 2. Then, by the Sobolev Embedding Theorem, A is of


class C1 so that PA is a bundle reduction of P of class C2. For any g ∈ Gk+1, letϑg denote the associated vertical automorphism of P , given by

ϑg(p) = p · g(p), ∀p ∈ P. (3)

For H ⊆ G, let CG(H) denote the centralizer in G. We abbreviate C2G(H) =

CG(CG(H)). Note that H ⊆ C2G(H).

Let A ∈ Ak. Since the elements of Gk+1A map A-horizontal paths in P to A-

horizontal paths, they are constant on PA. Conversely, any gauge transformationwhich is constant on PA leaves A invariant. Thus, for any g ∈ Gk+1 one has

g ∈ Gk+1A ⇐⇒ g|PA

is constant. (4)

This suggests characterizing orbit types by certain classes of bundle reductions ofP . These will be constructed now. For any subgroup S ⊆ Gk+1, define a subset�(S) ⊆ P by

�(S) = {p ∈ P | g(p) = g(p0) ∀g ∈ S}. (5)

LEMMA 3.1.

(a) For any A ∈ Ak, �(Gk+1A ) = PA · C2

G(HA).(b) Let A, A′ ∈ Ak, then �(Gk+1

A ) = �(Gk+1A′ ) implies Gk+1

A = Gk+1A′ .

(c) Let g ∈ Gk+1. For any subgroup S ⊆ Gk+1, �(gSg−1) = ϑg(�(S)) · g(p0)−1.

Remark. According to (a), if the subgroup S is the stabilizer of a connection A,then �(S) is a bundle reduction of P . In [23], the image PA ·C2

G(HA) is called theevolution bundle generated by A.

Proof. (a) Let A ∈ Ak. Recall that PA has a structure group HA. Hence, in viewof (4), the equivariance property of gauge transformations implies

{g(p0) | g ∈ Gk+1A } = CG(HA). (6)

Thus, by equivariance again,

g ∈ Gk+1A �⇒ g|PA·C2

G(HA) is constant. (7)

This shows PA · C2G(HA) ⊆ �(Gk+1

A ). Conversely, let p ∈ P such that g(p) =g(p0) for all g ∈ Gk+1

A . There exists a ∈ G such that p · a−1 ∈ PA. Due to (4),

g(p0) = g(p · a−1) = ag(p)a−1 = ag(p0)a−1, ∀g ∈ Gk+1

A .

Due to (6), then a ∈ C2G(HA). Hence, p = (p · a−1) · a ∈ PA · C2

G(HA).(b) Let A, A′ ∈ Ak be given. For any g ∈ Gk+1, we have

g ∈ Gk+1A ⇐⇒ g|�(Gk+1

A ) is constant.


Here implication from left to right is due to (7) and assertion (a), the inverse impli-cation follows from PA ⊆ �(Gk+1

A ) and (4). Since a similar characterization holdsfor Gk+1

A′ , the assertion follows.(c) Let p ∈ P , h ∈ S. Using (3) we compute

g(p0)−1g(p)h(p)g(p)−1g(p0) = h(ϑg−1(p) · g(p0)).

This allows us to write down the following chain of equivalences:

p ∈ �(gSg−1) ⇐⇒ g(p)h(p)g(p)−1 = g(p0)h(p0)g(p0)−1 ∀h ∈ S

⇐⇒ g(p0)−1g(p)h(p)g(p)−1g(p0) = h(p0) ∀h ∈ S

⇐⇒ h(ϑg−1(p) · g(p0)) = h(p0) ∀h ∈ S

⇐⇒ ϑg−1(p) · g(p0) ∈ �(S).

This proves assertion (c). ✷DEFINITION 3.2. A bundle reduction Q ⊆ P will be called holonomy-inducedof class Cr iff there exists a connected reduction Q ⊆ P to a subgroup H such that

Q = Q · C2G(H ). (8)

Let Red∗(P ) denote the set of isomorphy classes of holonomy-induced bundlereductions of P of class C0, factorized by the action of the structure group G.We equip Red∗(P ) with the following natural partial ordering: η � η′ iff thereexist representatives Q of η and Q′ of η′ such that Q ⊆ Q′.

It is evident that in the definition of Red∗(P ), continuity could be replaced byany differentiability class.

THEOREM 3.3. Let M be compact, dim M � 2. Then the assignment � induces,by passing to quotients, an order-preserving bijection from �k onto Red∗(P ).

Proof. Let τ ∈ �k and choose a representative S ⊆ Gk+1. There exists A ∈Ak such that S = Gk+1

A . According to Lemma 3.1(a), �(S) can be obtained byextending the bundle reduction PA ⊆ P to the structure group C2

G(HA). SincePA is of class C0, so is �(S). Since PA is connected, �(S) is holonomy-inducedof class C0. According to Lemma 3.1(c), if S is conjugate in Gk+1 to some S ′,�(S) and �(S ′) are conjugate under the actions of Gk+1 and G, then, since gaugetransformations from Gk+1 are continuous, �(S) and �(S ′) are C0-isomorphic.Thus, � projects to a map from �k to Red∗(P ).

To check that this map is surjective, let Q ⊆ P be a holonomy-induced bundlereduction of P of class C0. Let Q ⊆ Q be a connected bundle reduction of P

of class C0, with a structure group H , such that (8) holds. Due to well-knownsmoothing theorems [17, Ch. I, §4], we may assume that Q and Q are of class C∞.Moreover, up to the action of G, p0 ∈ Q. Since M is compact and dim M � 2,Q carries a C∞-connection with holonomy group H [21, Ch. II, Thm. 8.2]. This


connection prolongs to a unique (smooth) A ∈ Ak obeying PA = Q and HA = H .Then Lemma 3.1(a) and (8) imply �(Gk+1

A ) = Q · C2G(H ) = Q. This proves

surjectivity.To show that the projected map is injective, let τ, τ ′ ∈ �k. Choose representa-

tives S, S ′ and assume that �(S ′) and �(S) ·a are C0-isomorphic, for some a ∈ G.Then there exists a continuous gauge transformation g such that

�(S ′) = ϑg(�(S) · a). (9)

LEMMA 3.4. Let A ∈ Ak and let Q ⊆ P be a bundle reduction of class C∞. Ifthere exists a continuous gauge transformation h of P such that

�(Gk+1A ) = ϑh(Q), (10)

then h may be chosen from Gk+1.

Before proving the lemma, let us assume that it holds and finish the arguments.Again, due to smoothing theorems, �(S) is C0-isomorphic to some bundle reduc-tion Q ⊆ P of class C∞, i.e., there exists a continous gauge transformation h suchthat �(S) = ϑh(Q). Due to Lemma 3.4, we can choose h ∈ Gk+1. Moreover, dueto (9), �(S ′) = ϑgh(Q · a). By application of Lemma 3.4 again, we can achievegh ∈ Gk+1. This shows that we may assume, from the beginning, g ∈ Gk+1.

Now consider (9). Since

p0 ∈ �(S), ϑg(p0) · a = p0 · (g(p0)a) ∈ �(S ′).

Since also p0 ∈ �(S ′), g(p0)a is an element of the structure group of �(S ′). Then�(S ′) · (a−1g(p0)

−1) = �(S ′), so that (9) and Lemma 3.1(c) yield

�(S ′) = ϑg(�(S)) · g(p0)−1 = �(gSg−1).

Due to Lemma 3.1(b), then S ′ = gSg−1. This proves injectivity.

Proof of Lemma 3.4. Let A and Q be given. Under the assumption that (10)holds, �(Gk+1

A ) and Q have the same structure group H . There exist an opencovering {Ui} and local trivializations

ξi: Ui ×H → �(Gk+1A )|Ui

of �(Gk+1A ) and ηi : Ui ×H → Q|Ui

of Q.

These define local trivializations ξi , ηi : Ui ×G → P |Uiof P over {Ui}. Here ηi ,

ηi are of class C∞. As for ξi and ξi , we note that �(Gk+1A ) contains the holonomy

bundle PA. Since A is of class Wk , PA admits local cross-sections of class Wk+1

(cf. the proof of Lemma 1 in [21, Ch. II, §7.1]). Hence, ξi and ξi may be chosenfrom the class Wk+1.

Due to (10), the family {ϑh ◦ ηi} defines a local trivialization of class C0 of�(Gk+1

A ) over {Ui}. Hence, there exists a vertical automorphism ϑ ′ of class C0 of�(Gk+1

A ) such that ξi = ϑ ′ ◦ ϑh ◦ ηi , ∀i. By equivariant prolongation, ϑ ′ defines


a unique gauge transformation h′ of P of class C0. Since ϑh′ leaves �(Gk+1A )

invariant, �(Gk+1A ) = ϑh′h(Q). Thus, by possibly redefining h we may assume

that h′ = 1, i.e., that ϑ ′ is trivial. Then

ξi = ϑh ◦ ηi , ∀i. (11)

As we shall argue now, (11) implies h ∈ Gk+1. By definition, h ∈ Gk+1 iff the localrepresentatives hi = h ◦ ηi ◦ ι are of class Wk+1. Here ι denotes the embeddingUi → Ui ×G, x �→ (x, 1). Using

ηi (x, hi(x)) = ϑh ◦ ηi(x, 1), ∀x ∈ Ui,

we find that hi = pr2 ◦ η−1i ◦ ϑh ◦ ηi ◦ ι, where pr2 is the canonical projection

Ui ×G → G. Using (11) we obtain hi = pr2 ◦ η−1i ◦ ξi ◦ ι, ∀i. Here ξi is of class

Wk+1 and the other maps are of class C∞. Thus, according to the compositionrules of Sobolev mappings, hi is of class Wk+1. It follows h ∈ Gk+1. This provesLemma 3.4 and, therefore, Theorem 3.3. ✷

Remarks. (1) As an important consequence of Theorem 3.3, �k does not dependon k.

(2) Theorem 3.3 also shows that the notion of holonomy-induced bundle reduc-tion may be viewed as an abstract version of the notion of evolution subbundlegenerated by a connection, introduced in [23].

(3) General arguments show that Red∗(P ) is countable, see [23, §4.2]. Hence,so is �k. Countability of �k is a necessary condition for this set to define a strat-ification in the sense of [22]. It was first stated in Theorem 4.2.1 in [23]. In fact,the proof of this theorem already contains most of the arguments needed to proveTheorem 3.3. Unfortunately, although in the proof of Theorem 4.2.1 the authorsused that isomorphy of evolution subbundles implies conjugacy under the actionof Gk+1, they did not give an argument for that. Such an argument is provided byour Lemma 3.4.

(4) The geometric ideas behind the proof of Theorem 3.3 are also containedin [15, §2]. However, a rigorous proof was not given there.

In view of Theorem 3.3, we are left with the problem of determining the setRed∗(P ) together with its partial ordering. To begin with, we make the followingobservation. By construction, the structure group of a holonomy-induced reductionof P has the form H = CG(H ), for some H ⊆ H . Such subgroups are known asHowe subgroups in the literature, cf. [27]. They can equivalently be characterizedby the property H = C2

G(H).

DEFINITION 3.5. A reduction of P to a Howe subgroup of G will be called aHowe subbundle.

As remarked above, the class of Howe subbundles of P contains the class ofholonomy-induced reductions of P . Thus, we are lead to the following programme:


PROGRAMME

Step 1. Determination of the Howe subgroups of G. Since G-action on bundle re-ductions conjugates the structure group, classification up to conjugacy is sufficient.Step 2. Determination of the Howe subbundles of P up to isomorphy.Step 3. Specification of the Howe subbundles which are holonomy-induced.Step 4. Factorization by G-action.Step 5. Determination of the natural partial ordering.

In this paper, we perform steps 1–4 for the group G = SU(n). The determina-tion of the natural partial ordering can be found in [37].

4. The Howe Subgroups of SU(n)

Let Howe(SU(n)) denote the set of conjugacy classes of Howe subgroups of SU(n).In order to derive Howe(SU(n)), we consider SU(n) as a subset of Mn(C), theassociative algebra of complex (n× n)-matrices.

In the literature, it is customary to consider, instead of Howe subgroups, reduc-tive Howe dual pairs. A Howe dual pair is an ordered pair of subgroups (H1, H2)

of G such that

H1 = CG(H2), H2 = CG(H1).

The assignment H �→ (H, CG(H)) defines a one-to-one relation between Howesubgroups and Howe dual pairs. The pair is called reductive iff its members arereductive. In our case, this condition is automatically satisfied because SU(n) iscompact and Howe subgroups are always closed. Reductive Howe dual pairs playan important role in the representation theory of Lie groups, cf. [18]. Althoughfor SU(n) it is not necessary to go into the details of the classification theory ofreductive Howe dual pairs, we note that there exist, essentially, two methods. Oneapplies to the isometry groups of Hermitian spaces and uses the theory of Hermitianforms [27, 29, 31]. The other method applies to complex semisimple Lie algebrasand uses root space techniques [30].

Let K(n) denote the collection of pairs of sequences (of equal length) of positiveintegers

J = (k, m) = ((k1, . . . , kr), (m1, . . . , mr)), r = 1, 2, 3, . . . , n,

which obey

k ·m =r∑

i=1

kimi = n. (12)

For a given element J = (k, m) of K(n), let g denote the greatest common divisorof the members of m. Define m = (m1, . . . , mr ) by gmi = mi , ∀i. Moreover, for


any permutation σ of r elements, define σJ = (σk, σm). Any J ∈ K(n) generatesa decomposition

Cn = (Ck1 ⊗ C

m1)⊕ · · · ⊕ (Ckr ⊗ Cmr ) (13)

and an associated injective homomorphism

Mk1(C)× · · · ×Mkr(C) → Mn(C)

(14)(D1, . . . , Dr) �→ (D1 ⊗ 1m1)⊕ · · · ⊕ (Dr ⊗ 1mr

).

We denote the image of this homomorphism by MJ (C), its intersection with U(n)

by U(J ) and its intersection with SU(n) by SU(J ). Note that U(J ) is the image ofthe restriction of (14) to U(k1)× · · · × U(kr) ⊆ Mk1(C)× · · · ×Mkr

(C).

LEMMA 4.1. A subgroup of U(n) (resp. SU(n)) is Howe if and only if it is con-jugate, under the action of SU(n) by inner automorphisms, to U(J ) (resp. SU(J ))for some J ∈ K(n).

Proof. We give only the proof for SU(n). Let H be a Howe subgroup of SU(n).Then H = CSU(n)(K) = CMn(C)(K) ∩ SU(n) for some subgroup K ⊆ SU(n).Since K is ∗-invariant, so is M ′ := CMn(C)(K). Since M ′ also contains the unitmatrix, it is a unital ∗-subalgebra (or von Neumann algebra) of Mn(C). Thus, as abasic fact, M ′ is conjugate under SU(n)-action to MJ (C), for some J . Then H isconjugate in SU(n) to MJ (C) ∩ SU(n) = SU(J ).

Conversely, let J ∈ K(n). It suffices to show that SU(J ) is Howe. Consider thecentralizer M ′ := CMn(C)(MJ (C)). Since MJ (C) is a unital ∗-subalgebra, so is M ′.In particular, M ′ is spanned by the subset M ′ = M ′ ∩ SU(n) (which is a sub-group of SU(n)). Moreover, the Double Commutant Theorem yields CMn(C)(M) =MJ (C). Thus, we obtain

CSU(n)(M′) = CMn(C)(M

′) ∩ SU(n) = CMn(C)(M′) ∩ SU(n)

= MJ (C) ∩ SU(n) = SU(J ).

This shows that SU(J ) is Howe. ✷LEMMA 4.2. Let J, J ′ ∈ K(n). Then SU(J ) and SU(J ′) are conjugate under theaction of SU(n) by inner automorphisms if and only if there exists a permutationσ such that J ′ = σJ .

Proof. It suffices to check the assertion for the subalgebras MJ (C) and MJ ′(C)

of Mn(C). If a permutation σ exists, there exists T ∈ SU(n) mapping the factorsC

k′i ⊗ Cm′i of the decomposition (13), defined by J ′, identically onto the factors

Ckσ(i) ⊗ C

mσ(i) of the decomposition defined by J . Then MJ ′(C) = T −1MJ (C)T .Conversely, if MJ ′(C) = T −1MJ (C)T for some T ∈ SU(n), then MJ (C) andMJ ′(C) are isomorphic. Hence, k′ = σk for some permutation σ . Since T is anisomorphism of the representations

Mk1(C)× · · · ×Mkr(C)

J−→ Mn(C)


and

Mk1(C)× · · · ×Mkr(C)

σ−→ Mk′1(C)× · · · ×Mk′r (C)J ′−→ Mn(C),

where J , J ′ indicate the respective embeddings (14), it does not change the multi-plicities of the irreducible factors. Thus, m′ = σm. It follows J ′ = σJ . ✷

As a consequence of Lemma 4.2, we introduce an equivalence relation on theset K(n): J ∼ J ′ iff J ′ = σJ for some permutation σ . Let K(n) denote the set ofequivalence classes.

THEOREM 4.3. The assignment J �→ SU(J ) induces a bijection from K(n) ontoHowe(SU(n)).

Proof. According to Lemma 4.1, the assignment J �→ SU(J ) induces a surjec-tive map K(n) → Howe(SU(n)). Due to Lemma 4.2, this map projects to K(n)

and the projected map is injective. ✷This concludes the classification of Howe subgroups of SU(n), i.e., Step 1 of

our programme.

5. The Howe Subbundles of SU(n)-Bundles

In this section, let P be a principal SU(n)-bundle over M, dim M � 4. We aregoing to derive the Howe subbundles of P up to isomorphy. As we have seen above,we can restrict attention to the structure groups SU(J ), J ∈ K(n). Thus, let J ∈K(n) be fixed. Let Bun(M, SU(J )) denote the set of isomorphism classes of princi-pal SU(J )-bundles over M (where principal bundle isomorphisms are assumed tocommute with the structure group action and to project to the identical map on thebase space). Moreover, let Red(P, SU(J )) denote the set of isomorphism classesof reductions of P to the subgroup SU(J ) ⊆ SU(n).

We shall first derive a description of Bun(M, SU(J )) in terms of suitable char-acteristic classes and then give a characterization of the subset Red(P, SU(J )). Theclassification of Bun(M, SU(J )) will involve the construction of the Postnikovtower of the classifying space BSU(J ) up to level 5. For the convenience of thereader, the basics of this method will be briefly explained below. Note that in thesequel maps of topological spaces are always assumed to be continuous, withoutexplicitly stating this.

5.1. PRELIMINARIES

Universal Bundles and Classifying Spaces. Let G be a Lie group. As a basicfact in bundle theory, there exists a so-called universal G-bundle G ↪→ EG →BG with the following property: For any CW complex (hence, in particular, anymanifold) X the assignment

[X, BG] −→ Bun(X, G), f �→ f ∗EG, (15)


is a bijection [19]. Here [·, ·] means the set of homotopy classes of maps and f ∗denotes the pull-back of bundles. Both EG and BG can be realized as CW com-plexes. They are unique up to homotopy equivalence. BG is called the classifyingspace of G. The homotopy class of maps X → BG associated to P ∈ Bun(X, G)

by virtue of (15) is called the classifying map of P . We denote it by fP . Note that aprincipal G-bundle is universal iff its total space is contractible. As a consequence,the exact homotopy sequence of fibre spaces [8] implies for the homotopy groups

πi(G) ∼= πi+1(BG), i = 0, 1, 2, . . . . (16)

Associated Principal Bundles Defined by Homomorphisms. Let ϕ: G → G′ be aLie group homomorphism and let P ∈ Bun(X, G). By virtue of the action

G×G′ → G′, (a, a′) �→ ϕ(a)a′,

G′ becomes a left G-space and we have an associated bundle P [ϕ] = P ×G G′.P [ϕ] can be viewed as a principal bundle in an obvious way. One has the naturalbundle morphism

ψ : P → P [ϕ], p �→ [(p, 1G′)]. (17)

It obeys ψ(p · a) = ψ(p) · ϕ(a) and projects to the identical map on X.In the special case where ϕ is a Lie subgroup embedding, (17) is an embedding

of P onto a reduction of P [ϕ] to the subgroup (G, ϕ) of G′. Then P [ϕ] is theextension of P by G′. In this case, if no confusion about ϕ can arise, we shalloften write P [G′] instead of P [ϕ].

Classifying Maps Associated to Homomorphisms. Again, let ϕ: G → G′ be ahomomorphism. One can associate to ϕ a map Bϕ: BG → BG′ which is definedas the classifying map of the principal G′-bundle (EG)[ϕ] associated to the univer-sal G-bundle EG. It has the following functorial property: For ϕ: G → G′ andϕ′: G′ → G′′ there holds

B(ψ ◦ ϕ) = Bψ ◦ Bϕ. (18)

Using Bϕ, the classifying map of P [ϕ] can be expressed through that of P :

fP [ϕ] = Bϕ ◦ fP . (19)

We note that in the special case where ϕ is a normal Lie subgroup embedding, Bϕ

is a principal bundle

G′/G ↪→ BGBϕ−→ BG′. (20)

The classifying map of this bundle is Bp [6], where p: G′ → G′/G is the naturalprojection.


Characteristic Classes. Let G be a Lie group. Consider the cohomology ringH ∗(BG, π) of the classifying space with values in some Abelian group π . Forany P ∈ Bun(X, G), the homomorphism (fP )∗, induced on cohomology, mapsH ∗(BG, π) to H ∗(X, π). Therefore, given γ ∈ H ∗(BG, π), one can define a map

χγ : Bun(X, G) → H ∗(X, π), P �→ (fP )∗γ. (21)

This is called the characteristic class for G-bundles over X defined by γ . Byconstruction, one has the following universal property of characteristic classes: Letf : X → X′ be a map and let P ′ ∈ Bun(X′, G). Then

χγ (f ∗P ′) = f ∗χγ (P ′). (22)

Observe that if two bundles are isomorphic then their images under arbitrary char-acteristic classes coincide, whereas the converse, in general, does not hold. Thisis due to the fact that characteristic classes can control maps X → BG only onthe level of the homomorphisms induced on cohomology. In general, the latter donot give sufficient information on the homotopy properties of the maps. In certaincases, however, they do. For example, such cases are obtained by specifying G tobe U(1) or discrete, or by restricting X in dimension. In these cases there exist setsof characteristic classes which classify Bun(X, G).

Eilenberg–MacLane Spaces. Let π be a group and n a positive integer. An ar-cwise connected CW complex X is called an Eilenberg–MacLane space of typeK(π, n) iff πn(X) = π and πi(X) = 0 for i �= n. Eilenberg–MacLane spaces existfor any choice of π and n, provided π is commutative for n � 2. They are uniqueup to homotopy equivalence. The simplest example of an Eilenberg–MacLanespace is the 1-sphere S1, which is of type K(Z, 1). Two further examples, K(Z, 2)

and K(Zg, 1), are briefly discussed in the Appendix. Note that Eilenberg–MacLanespaces are, apart from very special examples, infinite dimensional.

Assume π to be commutative also in the case n = 1. Due to the Univer-sal Coefficient Theorem, Hom(Hn(K(π, n)), π) is isomorphic to a subgroup ofH n(K(π, n), π). Due to the Hurewicz Theorem, Hn(K(π, n)) ∼= πn(K(π, n)) =π . It follows that H n(K(π, n), π) contains elements which correspond to iso-morphisms Hn(K(π, n)) → π . Such elements are called characteristic. If γ ∈H n(K(π, n), π) is characteristic then for any CW complex X, the map

[X, K(π, n)] → H n(X, π), f �→ f ∗γ, (23)

is a bijection [8, §VII.12]. In this sense, Eilenberg–MacLane spaces provide a linkbetween homotopy properties and cohomology.

Path-Loop Fibration. Let X be an arcwise connected topological space. Considerthe path-loop fibration over X, 0(X) ↪→ P(X) −→ X. Here 0(X) and P(X)

denote the loop space and the path space of X, respectively (both based at some


point x0 ∈ X). Since P(X) is contractible, the exact homotopy sequence inducedby the path-loop fibration implies πi(0(X)) ∼= πi+1(X), i = 0, 1, 2, . . . . Thus,0(K(π, n + 1)) = K(π, n), ∀n, and the path-loop fibration over K(π, n + 1)

reads

K(π, n) ↪→ P(K(π, n+ 1)) → K(π, n+ 1). (24)

Postnikov Tower. A map f : X → X′ of topological spaces is called an n-equiv-alence iff the homomorphism induced on homotopy groups f∗: πi(X) → πi(X

′)is an isomorphism for i < n and surjective for i = n. Let f : X → X′ be ann-equivalence and let Y be a CW complex. Then the map [Y, X] → [Y, X′],g �→ f ◦ g, is bijective for dim Y < n and surjective for dim Y = n [8, Ch. VII,Cor. 11.13].

A CW complex Y is called simple iff it is arcwise connected and the natural ac-tion of π1(Y ) on πi(Y ) is trivial for all i � 1. The following theorem describes howa simple CW complex can be approximated by n-equivalent spaces constructedfrom Eilenberg–MacLane spaces.

THEOREM 5.1. Let Y be a simple CW complex. There exist:

(a) a sequence of CW complexes Yn and principal fibrations

K(πn(Y ), n) ↪→ Yn+1qn−→ Yn, n = 1, 2, 3, . . . , (25)

induced by maps θn: Yn → K(πn(Y ), n+ 1),(b) a sequence of n-equivalences yn: Y → Yn, n = 1, 2, 3, . . . ,

such that Y1 = ∗ (one point space) and qn ◦ yn+1 = yn for all n.

Proof. The assumption that Y be simple implies that the constant map Y → ∗is a simple map (see [8, Ch. VII, Def. 13.4] for a definition of the latter). Thus, theassertion is a consequence of a more general theorem about simple maps given in[8, Ch. VII, Thm. 13.7]. ✷

Remarks. (1) The sequence of spaces and maps (Yn, yn, qn), n = 1, 2, 3, . . . , iscalled Postnikov tower, or Postnikov system, or Postnikov decomposition of Y .

(2) For the principal fibrations (25) to be induced by a map θn: Yn → K(πn(Y ),

n + 1) means that they are given as pull-back of the path-loop fibration (24) overK(πn(Y ), n+ 1).

Strategy. We wish to classify Bun(M, SU(J )) by means of characteristic classes.For that purpose, we have to find out whether this is possible and which char-acteristic classes are necessary for classification. We start from the general clas-sification result Bun(M, SU(J )) = [M, BSU(J )]. In general, [M, BSU(J )] ishard to handle and it cannot be expected to be classified by characteristic classes.However, Theorem 5.1 allows us to successively construct n-equivalent approxi-mations BSU(J )n, up to n = 5, starting from BSU(J )1 = ∗. Thus, if we assume


dim M � 4, [M, BSU(J )] = [M, BSU(J )5] and the explicit form of BSU(J )

allows us to determine the kind of characteristic classes which are necessary toclassify Bun(M, SU(J )). Finally, we shall construct these classes explicitly.

We remark that the strategy described is usual in dealing with bundle classifica-tion problems, see, for instance, [5, 36].

Now let us turn to the construction of BSU(J5). First of all, we need informationabout the low-dimensional homotopy groups of SU(J ).

5.2. THE HOMOTOPY GROUPS OF SU(J )

For a positive integer a, denote the embedding Za ↪→ U(1) by ja and the endomor-phism of U(1) mapping z �→ za by pa . Let jJ and iJ denote the natural embeddingsSU(J ) ↪→ U(J ) and U(J ) ↪→ U(n), respectively. Finally, let prU(J )

i : U(J ) →U(ki) denote the natural projections.

Recall that SU(J ) = ker(detU(n) ◦ iJ ). Let D ∈ U(J ). Writing D = (D1 ⊗1m1)⊕ · · · ⊕ (Dr ⊗ 1mr

), where Di = prU(J )i (D) ∈ U(ki), we have

detU(n) ◦ iJ (D) =r∏

i=1

pmi◦ detU(ki)(Di) = pg

(r∏

i=1

pmi◦ detU(ki)(Di)

).

Thus, we can decompose

detU(n) ◦ iJ = pg ◦ λJ , (26)

where λJ : U(J ) → U(1) is defined by

λJ (D) =r∏

i=1

pmi◦ detU(ki) ◦ prU(J )

i (D), ∀D ∈ U(J ). (27)

Due to (26), the restriction of λJ to the subgroup SU(J ) takes values in ker pg =jg(Zg). Hence, we can define λS

J : SU(J ) → Zg by requiring

λJ ◦ jJ = jg ◦ λSJ . (28)

In the following lemma, let (SU(J ))0 denote the arcwise connected componentof the identity. Note that it is also a connected component.

LEMMA 5.2. The homomorphism λSJ projects to an isomorphism SU(J )/

(SU(J ))0 → Zg.Proof. Consider the homomorphism λS

J : SU(J ) → Zg. The target space be-ing discrete, λS

J must be constant on connected components. Hence, (SU(J ))0 ⊆ker λS

J , so that λSJ projects to a homomorphism SU(J )/(SU(J ))0 → Zg. The latter

is surjective, because λSJ is surjective. To prove injectivity, we show ker λS

J ⊆(SU(J ))0. Let D ∈ ker λS

J and denote Di = prU(J )i ◦ jJ (D). Define

ϕ: U(1r ) → U(1), (z1, . . . , zr) �→ zm11 · · · zmr

r .


Then λSJ (D) = ϕ(detU(k1) D1, . . . , detU(kr ) Dr). By assumption, (detU(k1) D1,

. . . , detU(kr ) Dr) ∈ ker ϕ. Since the exponents defining ϕ have greatest common di-visor 1, ker ϕ is connected. Thus, there exists a path (γ1(t), . . . , γr(t)) in ker ϕ run-ning from (detU(k1) D1, . . . , detU(kr) Dr) to (1, . . . , 1). For each i = 1, . . . , r, definea path Gi(t) in U(ki) as follows: First, go from Di to (detU(ki) Di)⊕ 1ki−1, keep-ing the determinant constant, thus using connectedness of SU(ki). Next, use thepath γi(t)⊕ 1ki−1 to get to 1ki

. By construction, the image of (G1(t), . . . , Gr(t))

under the embedding (14) is a path in SU(J ) connecting D with 1n. This provesker λS

J ⊆ (SU(J ))0. ✷THEOREM 5.3. The homotopy groups of SU(J ) are

π0(SU(J )) ∼= Zg, π1(SU(J )) ∼= Z⊕(r−1)

and

πi(SU(J )) ∼= πi(U(k1))⊕ · · · ⊕ πi(U(kr)) for i > 1.

In particular, π1(SU(J )) and π3(SU(J )) are torsion-free.Proof. The group π0(SU(J )) = SU(J )/(SU(J ))0 is given by Lemma 5.2. For

i > 1, the assertion follows from the exact homotopy sequence induced by thebundle SU(J ) ↪→ U(J ) → U(1) with projection ϕ = detU(n) ◦ iJ . For i = 1,consider the following portion of this sequence:

π2(U(1))→ π1(SU(J ))→ π1(U(J ))ϕ∗→ π1(U(1))→ π0(SU(J ))→ π0(U(J ))

0 → π1(SU(J ))→ Z⊕r → Z → Zg → 0.

One has Z⊕r/ ker(ϕ∗) ∼= im (ϕ∗). Exactness implies

ker(ϕ∗) ∼= π1(SU(J )) and im (ϕ∗) = gZ ∼= Z.

It follows π1(SU(J )) ∼= Z⊕(r−1), as asserted. ✷

5.3. THE POSTNIKOV TOWER OF BSU(J ) UP TO LEVEL 5

Let r∗ denote the number of indices i for which ki > 1.

THEOREM 5.4. The fifth level of the Postnikov tower of BSU(J ) is given by

(BSU(J ))5 = K(Zg, 1)×r−1∏j=1

K(Z, 2)×r∗∏

j=1

K(Z, 4). (29)


Proof. First, we check that BSU(J ) is a simple space. To see this, note thatany inner automorphism of SU(J ) is generated by an element of (SU(J ))0, henceis homotopic to the identity automorphism. Consequently, the natural action ofπ0(SU(J )) on πi−1(SU(J )), i = 1, 2, 3, . . . , induced by inner automorphisms, istrivial. Since the natural isomorphisms πi−1(SU(J )) ∼= πi(BSU(J )) transform thisaction into that of π1(BSU(J )) on πi(BSU(J )), the latter is trivial, too. Thus, wecan apply Theorem 5.1 to construct the Postnikov tower of BSU(J ). According toTheorem 5.3, the relevant homotopy groups are

π1(BSU(J )) = Zg, π2(BSU(J )) = Z⊕(r−1),

(30)π3(BSU(J )) = 0, π4(BSU(J )) = Z

⊕r∗.

Moreover, we shall need that H ∗(K(Z, 2), Z) is torsion-free and that

H 2i+1(K(Z, 2), Z) = 0,

H 2i+1(K(Zg, 1), Z) = 0, i = 0, 1, 2, . . . ,(31)

see Appendix. We start with (BSU(J ))1 = ∗.

(BSU(J ))2: Being a fibration over (BSU(J ))1, (BSU(J ))2 must coincide with thefibre:

(BSU(J ))2 = K(Zg, 1). (32)

(BSU(J ))3: In view of (32) and (30), (BSU(J ))3 is the total space of a fibration

K(Z⊕(r−1), 2) ↪→ (BSU(J ))3q2−→ K(Zg, 1) (33)

induced from the path-loop fibration over K(Z⊕(r−1), 3) by somemap θ2: K(Zg, 1) → K(Z⊕(r−1), 3). Note that K(Z⊕(r−1), n) =∏r−1

j=1 K(Z, n), ∀n. Then, due to (23),

[K(Zg, 1), K(Z⊕(r−1), 3)] =r−1∏i=1

H 3(K(Zg, 1), Z).

Here the right-hand side is trivial by (31). Hence, θ2 is homotopic toa constant map, so that the fibration (33) is trivial. It follows that

(BSU(J ))3 = K(Zg, 1)×r−1∏j=1

K(Z, 2). (34)

(BSU(J ))4: In view of (30), (BSU(J ))4 is given by a fibration over (BSU(J ))3

with fibre K(0, 3) = ∗. Hence, it just coincides with the base space.


(BSU(J ))5: This is the total space of a fibration

K(Z⊕r∗, 4) ↪→ (BSU(J ))5q4→ (BSU(J ))3, (35)

which is induced by a map θ4: (BSU(J ))3 → K(Z⊕r∗, 5). Similarlyto the case of θ2,

[(BSU(J ))3, K(Z⊕r∗, 5)] =r∗∏

i=1

H 5((BSU(J ))3). (36)

Now consider (33). Since H ∗(K(Z, 2), Z) is torsion-free, we can apply the Kün-neth Theorem for cohomology [25, Ch. XIII, Cor. 11.3] to write H 5((BSU(J ))3)

as a sum over tensor products

H j(K(Zg, 1), Z)⊗H j1(K(Z, 2), Z)⊗ · · · ⊗H jr−1(K(Z, 2), Z),

where j + j1 + · · · + jr−1 = 5. Due to this constraint, each summand contains atensor factor of odd degree, hence is trivial by (31). Then (36) is trivial, and so isthe fibration (35). This proves the assertion. ✷

The fact that (BSU(J ))5 is a direct product of Eilenberg–MacLane spaces im-mediately yields the following corollary.

COROLLARY 5.5. Let J ∈ K(n) and dim M � 4. Let P, P ′ ∈ Bun(M, SU(J )).Assume that for any characteristic class α defined by an element of H 1(BSU(J ),

Zg), H 2(BSU(J ), Z), or H 4(BSU(J ), Z) there holds α(P ) = α(P ′). Then P andP ′ are isomorphic.

Proof. Let pr1, pr21, . . . , pr2r-1, and pr41, . . . , pr4r∗ denote the natural projec-tions of the direct product (29) onto its factors. Let γ1, γ2, and γ4 be characteris-tic elements of H 1(K(Zg, 1), Zg), H 2(K(Z, 2), Z), and H 4(K(Z, 4), Z), respec-tively. Consider the map

ϕ: [M, BSU(J )] → [M, (BSU(J ))5]

→ [M, K(Zg, 1)] ×r−1∏i=1

[M, K(Z, 2)] ×r∗∏

i=1

[M, K(Z, 4)]

→ H 1(M, Zg)×r−1∏i=1

H 2(M, Z)×r∗∏

i=1

H 4(M, Z),

that takes f to

(f ∗(pr1 ◦ y5)∗γ1, {f ∗(pr2i ◦ y5)

∗γ2}r−1i=1 , {f ∗(pr4i ◦ y5)

∗γ4}r∗i=1).

Here y5: BSU(J ) → (BSU(J ))5 is the 5-equivalence provided by Theorem 5.1.According to Theorem 5.4, the second step of ϕ and, therefore, the whole map, isa bijection.


Now let P, P ′ ∈ Bun(M, SU(J )) as proposed in the corollary. Then, by as-sumption, the homomorphisms (fP )∗ and (fP ′)

∗, induced on H 1(BSU(J ), Zg),H 2(BSU(J ), Z), and H 4(BSU(J ), Z), coincide. This implies ϕ(fP ) = ϕ(fP ′).Hence, fP and fP ′ are homotopic. This proves the corollary. ✷

We remark that, of course, the cohomology elements

(pr1 ◦ y5)∗γ1, (pr2i ◦ y5)

∗γ2, i = 1, . . . , r − 1, and

(pr4i ◦ y5)∗γ4, i = 1, . . . , r∗

define a set of characteristic classes which classifies Bun(M, SU(J )). These classesare independent and surjective. However, they are hard to handle, because we donot know the homomorphism y∗5 explicitly. Therefore, we prefer to work with char-acteristic classes defined by some natural generators of the cohomology groups inquestion. The price we have to pay for this is that the classes so constructed aresubject to a relation and that we have to determine their image explicitly.

5.4. GENERATORS FOR H ∗(BSU(J ), Z)

Instead of generators for H 2(BSU(J ), Z) and H 4(BSU(J ), Z) only, we can con-struct generators for the whole of H ∗(BSU(J ), Z) without any additional effort.Consider the homomorphisms

BSU(J )BjJ−→ BU(J )

BprU(J )i−→ BU(ki).

Recall that H ∗(BU(k), Z) is generated freely over Z by the elements

γ(2j)

Uk ∈ H 2j (BU(k), Z), j = 1, . . . , k,

see [6]. We assume that the signs of the γ(2j)

U(ki)are chosen in such a way that for the

canonical embedding ϕ: U(k)→ U(l), k � l, one has ϕ∗γ (2j)

U(l) = γ(2j)

U(k), 0 � j � k.

Then the characteristic class defined by γ(2j)

U(k) is the j th Chern class of U(k)-bundlesover M. We denote

γU(k) = 1+ γ(2)

U(k) + · · · + γ(2k)

U(k). (37)

Of course, γU(k) defines the total Chern class. The generators γ(2j)

U(ki)define elements

γ(2j)

J,i = (B prU(J )i )∗γ (2j)

U(ki), (38)

γ(2j)

J,i = (BjJ )∗(B prU(J )i )∗γ (2j)

U(ki)(39)

of H 2j (BU(J ), Z) and H 2j (BSU(J ), Z), respectively. We denote

γJ,i = 1+ γ(2)J,i + · · · + γ

(2ki)J,i , i = 1, . . . , r, (40)

γJ,i = 1+ γ(2)J,i + · · · + γ

(2ki)J,i , i = 1, . . . , r, (41)

as well as γJ = (γJ,1, . . . , γJ,r ) and γJ = (γJ,1, . . . , γJ,r).


LEMMA 5.6. H ∗(BU(J ), Z) is generated freely over Z by γ(2j)

J,i , j = 1, . . . , ki ,i = 1, . . . , r.

Proof. Using the isomorphism

U(J )dr−→

r∏i=1

U(J )<r

i=1prU(J )i−→

r∏i=1

U(ki),

where dr denotes r-fold diagonal embedding, the assertion follows from the factthat H ∗(<iBU(ki), Z) is generated freely over Z by elements 1BU(k1) × · · · ×1BU(ki−1) × γ

(2j)

U(ki)× 1BU(ki+1) × · · · × 1BU(kr ), j = 1, . . . , ki , i = 1, . . . , r. Here

× stands for the cohomology cross-product, and 1BU(ki) denotes the generator ofH 0(BU(ki), Z). ✷LEMMA 5.7. (BjJ )∗ is surjective.

Proof. According to (20) and due to U(J )/SU(J ) ∼= U(1), BjJ is the projectionin a principal bundle U(1) ↪→ BSU(J ) → BU(J ). Denote this bundle by η. Dueto π1(BU(J )) ∼= π0(U(J )) = 0, η is orientable, see [9, Def. 7.3.3]. Therefore, itinduces a Gysin sequence, see [9, §7.3.1],

H 1(BU(J ), Z)

(BjJ )∗−→ H 1(BSU(J ), Z)σ ∗−→ H 0(BU(J ), Z)

=c1(η)−→ H 2(BU(J ), Z)

(BjJ )∗−→ H 2(BSU(J ), Z)σ ∗−→ H 1(BU(J ), Z)

=c1(η)−→ H 3(BU(J ), Z) −→ · · · . (42)

(On the level of differential forms, the homomorphism σ ∗ is given by integrationover the fibre.) If η was trivial, we would have π1(BSU(J )) ∼= π1(BU(J ) ×U(1)) ∼= Z, which would contradict Theorem 5.3. Hence, η is nontrivial, so thatc1(η) �= 0. Due to Lemma 5.6, H ∗(BU(J ), Z) has no zero divisors. It follows thatmultiplication by c1(η) is an injective operation on H ∗(BU(J ), Z). Then exact-ness of the Gysin sequence (42) implies that the homomorphism σ ∗ is trivial and,therefore, (BjJ )∗ is surjective. ✷

Lemmas 5.6 and 5.7 yield the following corollary:

COROLLARY 5.8. H ∗(BSU(J ), Z) is generated over Z by γ(2j)

J,i , j = 1, . . . , ki ,i = 1, . . . , r.

Remark. The generators γ(2)J,i of H ∗(BSU(J ), Z) are subject to a relation. Since

this relation turns out to be a consequence of a more fundamental relation whichwill be derived below, it does not play a role in the sequel.


5.5. GENERATOR FOR H 1(BSU(J ), Zg)

Consider the homomorphism λSJ : SU(J ) → Zg and the induced homomorphism

(BλSJ )∗: H 1(BZg, Zg) −→ H 1(BSU(J ), Zg). (43)

Due to Lemma 5.2, λSJ ∗: π0(SU(J )) → π0(Zg) is an isomorphism. Hence, so is

(BλSJ )∗: π1(BSU(J )) → π1(BZg). Then the Hurewicz and Universal Coefficient

Theorems imply that (43) is an isomorphism. Thus, generators of H 1(BSU(J ), Zg)

can be obtained as the images of generators of H 1(BZg, Zg) under (BλSJ )∗. Note

that H 1(BZg, Zg) ∼= Zg, see Appendix. To choose a generator, we recall that theshort exact sequence

0 −→ Zµg−→ Z

@g−→ Zg −→ 0, (44)

where µg denotes multiplication by g and @g reduction modulo g, induces a longexact sequence of coefficient homomorphisms, see [8, §IV.5],

· · · βg−→ H i(·, Z)µg−→ H i(·, Z)

@g−→ H i(·, Zg)βg−→ H i+1(·, Z)

µg−→ · · · . (45)

The connecting homomorphism βg is usually called Bockstein homomorphism.

LEMMA 5.9. There exists a unique element δg ∈ H 1(BZg, Zg) such that

βg(δg) = (Bjg)∗γ (2)

U(1). (46)

It is a generator of H 1(BZg, Zg).Proof. First we notice that both βg(δg) and (Bjg)

∗γ (2)

U(1) are elements ofH 2(BZg, Zg) so that Equation (46) makes sense. Consider the following portionof the exact sequence (45):

H 1(BZg, Z)@g−→ H 1(BZg, Zg)

βg−→ H 2(BZg, Z)µg−→ H 2(BZg, Z)

0 −→ Zg −→ Zg −→ Zg.

(See Appendix for the cohomology groups.) Since µg is trivial here, βg is anisomorphism. Hence, we can define δg = β−1

g ◦ (Bjg)∗γ (2)

U(1). In order to checkthat this is a generator, consider J ◦ = ((1), (g)) ∈ K(g). Observe that Zg

∼=SU(J ◦), U(1) ∼= U(J ◦), and that jg corresponds to jJ ◦ : SU(J ◦) → U(J ◦). ThenLemma 5.7 implies that (Bjg)

∗ is surjective. Thus, H 2(BZg, Z) is generated by(Bjg)

∗γ (2)

U(1) and, therefore, H 1(BZg, Zg) is generated by δg. ✷We define δJ = (BλS

J )∗δg.

COROLLARY 5.10. H 1(BSU(J ), Zg) is generated by δJ , where βg(δJ ) =(BλS

J )∗(Bjg)∗γ (2)

U(1).


5.6. THE RELATION BETWEEN GENERATORS

Since βg maps H 1(BSU(J ), Zg) to H 2(BSU(J ), Z), it establishes a relation be-tween the generators of these groups. This will be derived now.

For any topological space X, let H even0 (X, Z) denote the subset of H even(X, Z)

consisting of elements of the form 1+ α(2)+α(4)+ · · · . It is a semigroup w.r.t. thecup product. Thus, for any finite sequence of nonnegative integers a = (a1, . . . , as),we can define a polynomial function

Ea:s∏

i=1

H even0 (X, Z) → H even

0 (X, Z), (α1, . . . , αs) �→ αa11 = . . . = αas

s , (47)

where powers are taken w.r.t. the cup product. By straightforward computation, forthe components of 2nd and 4th degree one obtains

E(2)a (α1, . . . , αs) =

s∑i=1

aiα(2)i , (48)

E(4)a (α1, . . . , αs) =

s∑i=1

aiα(4)i +

s∑i=1

ai(ai − 1)

2α

(2)i = α

(2)i +

+s∑

i<j=2

aiaj α(2)i = α

(2)j . (49)

As an immediate consequence of (48), for any l ∈ Z,

E(2)l a = lE(2)

a . (50)

LEMMA 5.11. The following two formulae hold:

(BiJ )∗γU(n) = Em(γJ ), (51)

(BλJ )∗γ (2)

U(1) = E(2)m (γJ ). (52)

Proof. First, consider (51). We decompose iJ as follows:

iJ : U(J )dr−→

∏i

U(J )<iprU(J )

i−→∏

i

U(ki)<idmi−→

∏i

(U(ki)mi× · · · × U(ki))

j−→ U(n).

Here dr , dmidenote r-fold and mi-fold diagonal embedding, respectively, and j

stands for the natural (blockwise) embedding. Since we have chosen the generatorsγ

(2j)

U(k) for different k in a consistent way,

(Bj)∗γU(n) = (γU(k1)

m1× · · · × γU(k1))× · · · × (γU(kr )

mr× · · · × γU(kr)).


Using this we obtain

(BiJ )∗γU(n) = d∗r ◦(∏

i

B prU(J )i

)∗◦(∏

i

dmi

)∗◦ (Bj)∗γU(n)

= d∗r ◦(∏

i

B prU(J )i

)∗◦(∏

i

dmi

)∗×

× ((γU(k1)

m1× · · · × γU(k1))× · · · × (γU(kr)

mr× · · · × γU(kr )))

= d∗r ◦(∏

i

B prU(J )i

)∗(γ

m1U(k1) × · · · × γ

mr

U(kr ))

= d∗r (γm1J,1 × · · · × γ

mr

J,r )

= γm1J,1 = . . . = γ

mr

J,r ,

hence (51). Now consider (52). Due to (26),

(BλJ )∗(Bpg)∗γ (2)

U(1) = (BiJ )∗(B detU(n))∗γ (2)

U(1).

Inserting

(B detU(n))∗γ (2)

U(1) = γ(2)

U(n) and (Bpg)∗γ (2)

U(1) = gγ(2)

U(1)

and using (51) and (50), we obtain

g(BλJ )∗γ (2)

U(1) = E(2)m (γJ ) = gE

(2)m (γJ ).

Since this relation holds in H 2(BU(J ), Z) which is free Abelian, it im-plies (52). ✷THEOREM 5.12. There holds the relation βg(δJ ) = E

(2)m (γJ ).

Proof. We compute

βg(δJ ) = (BλSJ )∗(Bjg)

∗γ (2)

U(1) by Corollary 5.10

= (BjJ )∗(BλJ )∗γ (2)

U(1) by (28)

= (BjJ )∗E(2)m (γJ ) by (52)

= E(2)m (γJ ). ✷

5.7. CHARACTERISTIC CLASSES FOR SU(J )-BUNDLES

Using the cohomology elements γ(2j)

J,i and δJ constructed above, we define thefollowing characteristic classes for SU(J )-bundles over a manifold M:

αJ,i: Bun(M, SU(J )) → H even0 (M, Z), Q �→ (fQ)∗γJ,i, i = 1, . . . , r, (53)

ξJ : Bun(M, SU(J )) → H 1(M, Zg), Q �→ (fQ)∗δJ . (54)


Sorted by degree,

αJ,i(Q) = 1+ α(2)J,i (Q)+ · · · + α

(2ki)J,i (Q),

where α(2j)

J,i (Q) = (fQ)∗γ (2j)

J,i . Moreover, we introduce the notation αJ (Q) =(αJ,1(Q), . . . , αJ,r(Q)). Then

αJ (Q) = (fQ)∗γJ (55)

and αJ can be viewed as a map from Bun(M, SU(J )) to the set

H (J)(M, Z)

={

(α1, . . . , αr) ∈r∏

i=1

H even0 (M, Z)

∣∣∣∣∣α(2j)

i = 0 for j > ki

}. (56)

By construction, the relation which holds for γJ and δJ carries over to the charac-teristic classes αJ and ξJ . From Theorem 5.12 we infer

E(2)m (αJ (Q)) = βg(ξJ (Q)), ∀Q ∈ Bun(M, SU(J )). (57)

In order to derive expressions for αJ and ξJ in terms of the ordinary characteristicclasses for U(ki)-bundles and Zg-bundles, let Q ∈ Bun(M, SU(J )). There are twokinds of principal bundles associated in a natural way to Q: The U(ki)-bundles

Q[prU(J )i ◦jJ ], i = 1, . . . , r, and the Zg-bundle Q[λS

J ]. For the first ones, using (19)and (39), we compute

c(Q[prU(J )i ◦jJ ]) = (f

Q[prU(J )

i◦jJ ])

∗γU(ki) = (fQ)∗ ◦ (BjJ )∗ ◦ (B prU(J )i )∗γU(ki)

= (fQ)∗γJ,i,

so that

αJ,i(Q) = c(Q[prU(J )i ◦jJ ]), i = 1, . . . , r. (58)

As for the second one, let χg denote the characteristic class for Zg-bundles overM defined by the generator δg ∈ H 1(BZg, Zg), i.e., χg(R) = (fR)∗δg , ∀R ∈Bun(M, Zg). Then (19) yields

χg(Q[λSJ ]) = (f

Q[λS

J])∗δg = (fQ)∗ ◦ (BλS

J )∗δg = (fQ)∗δJ .

Consequently,

ξJ (Q) = χg(Q[λSJ ]). (59)

5.8. CLASSIFICATION OF SU(J )-BUNDLES

We denote

K(M, J ) = {(α, ξ) ∈ H (J)(M, Z)×H 1(M, Zg) | E(2)m (α) = βg(ξ)}.


THEOREM 5.13. Let M be a manifold, dim M � 4, and let J ∈ K(n). Thenthe characteristic classes αJ and ξJ define a bijection from Bun(M, SU(J )) ontoK(M, J ).

Proof. The map is injective by Corollary 5.5. In the following lemma we provethat it is also surjective. ✷LEMMA 5.14. Let M be a manifold, dim M � 4, and let J ∈ K(n). Let (α, ξ) ∈K(M, J ). Then there exists Q ∈ Bun(M, SU(J )) such that αJ (Q) = α andξJ (Q) = ξ .

Proof. We give a construction of Q in terms of U(ki) and Zg-bundles. Thereexists R ∈ Bun(M, Zg) such that χg(R) = ξ . Due to dim M � 4, there exist alsoQi ∈ Bun(M, U(ki)) such that c(Qi) = αi , i = 1, . . . , r. Define Q = Q1 ×M

· · · ×M Qr (Whitney, or fibre, product). By identifying U(k1) × · · · × U(kr) withU(J ), Q becomes a U(J )-bundle. Then

Q[prU(J )i ] ∼= Qi, i = 1, . . . , r. (60)

Consider the U(1)-bundles Q[λJ ] and R[jg] associated to Q and R, respectively.Assume, for a moment, that they are isomorphic. Then R is a reduction of Q[λJ ]with structure group Zg. Let Q denote the pre-image of R under the natural bundlemorphism Q → Q[λJ ], see (17). This is a reduction of Q with structure groupbeing the pre-image of Zg under λJ , i.e., with structure group SU(J ). Using (58),Q[jJ ] = Q, and (60), for i = 1, . . . , r,

αJ,i(Q) = c(Q[prU(J )i ◦jJ ]) = c((Q[jJ ])[prU(J )

i ]) = c(Q[prU(J )i ]) = c(Qi) = αi.

Moreover, by construction of Q, Q[λSJ ] ∼= R. Thus, (59) yields ξJ (Q) =

χg(R) = ξ .It remains to prove Q[λJ ] ∼= R[jg]. Using (19), (52), (38) and (60) we find

c1(Q[λJ ]) = E

(2)m (α). (61)

Using (19) and (46) we get

c1(R[jg]) = βg(ξ). (62)

Thus, due to (α, ξ) ∈ K(M, J ), c1(Q[λJ ]) = c1(R

[jg]). It follows that, indeed,Q[λJ ] ∼= R[jg]. This proves the lemma and, therefore, the theorem. ✷

5.9. CLASSIFICATION OF SU(J )-SUBBUNDLES OF SU(n)-BUNDLES

Let P be a principal SU(n)-bundle over a manifold M and let J ∈ K(n). We aregoing to characterize the subset Red(P, SU(J )) ⊆ Bun(M, SU(J )) in terms ofthe characteristic classes αJ and ξJ . Recall that for Q ∈ Bun(M, SU(J )), Q[SU(n)]denotes the extension of Q by SU(n).


LEMMA 5.15. For any Q ∈ Bun(M, SU(J )), c(Q[SU(n)]) = Em(αJ (Q)).Proof. Note that c(Q[SU(n)]) = c(Q[U(n)]) = c(Q[iJ ◦jJ ]). Hence, using (19), (51),

(39) and (55) we obtain c(Q[SU(n)]) = Em(αJ (Q)). ✷We define K(P, J ) = {(α, ξ) ∈ K(M, J ) | Em(α) = c(P )}.

THEOREM 5.16. Let P be a principal SU(n)-bundle over a manifold M,dim M � 4, and let J ∈ K(n). Then the characteristic classes αJ , ξJ define abijection from Red(P, SU(J )) onto K(P, J ).

Proof. Let Q ∈ Bun(M, SU(J )). Then (αJ (Q), ξJ (Q)) ∈ K(M, J ). Lem-ma 5.15 implies that (αJ (Q), ξJ (Q)) ∈ K(P, J ) if and only if c(Q[SU(n)]) =c(P ). Due to dim M � 4, the latter is equivalent to Q[SU(n)] ∼= P , i.e., to Q ∈Red(P, SU(J )). ✷

The equation Em(α) = c(P ) actually contains the two equations E(2)m (α) = 0

and E(4)m (α) = c2(P ). However, under the assumption that (α, ξ) ∈ K(M, J ), the

first one is redundant, because then, due to (50),

E(2)m (α) = gE

(2)m (α) = gβg(ξ) = 0.

Thus, the relevant equations are

E(2)m (α) = βg(ξ), (63)

E(4)m (α) = c2(P ), (64)

where α ∈ H (J)(M, Z), ξ ∈ H 1(M, Zg). The set of solutions of Equation (63)yields K(M, J ), hence Bun(M, SU(J )). The set of solutions of both Equations (63)and (64) yields K(P, J ) and, therefore, Red(P, SU(J )).

This concludes the classification of Howe subbundles of P , i.e., Step 2 of ourprogramme.

5.10. EXAMPLES

We are going to determine K(P, J ) for several choices of J and for base manifoldsM = S4, S2 × S2, T4, and L3

p × S1. Here L3p denotes the three-dimensional lens

space which is defined to be the quotient of the restriction of the natural actionof U(1) on the sphere S3 ⊂ C

2 to the subgroup Zp. (Note that there exist moregeneral lens spaces in three dimensions.)

Preliminary Remarks. Due to compactness and orientability, H 4(M, Z) ∼= Z.Let us derive the Bockstein homomorphism βg: H 1(M, Zg) → H 2(M, Z). Sincefor products of spheres the integer-valued second cohomology is torsion-free, βg

is trivial here. For M = L3p × S1, we use the Universal Coefficient Theorem


to compute the necessary cohomology groups of L3p from its singular homology

which can be found in most textbooks, see, for example, [35, §II.7.7]:

H 1(L3p, Z) = 0, H 2(L3

p, Z) = Zp, H 1(L3p, Zg) = Z〈p,g〉. (65)

Here 〈p, g〉 denotes the greatest common divisor of p and g. Let 1L3p;Zg

, γ(1)

L3p;Zg

,

and γ(2)

L3p;Z be generators of H 0(L3

p, Zg), H 1(L3p, Zg), and H 2(L3

p, Z), respectively.

Moreover, we choose generators 1S1 and γ(1)

S1 of H 0(S1, Z) and H 1(S1, Z). Accord-ing to the Künneth Theorem for cohomology,

H 1(L3p × S1, Zg) = Z〈p,g〉 ⊕ Zg, H 2(L3

p × S1, Z) = Zp, (66)

with generators γ(1)

L3p;Zg×1S1 and 1L3

p;Zg×γ

(1)

S1 in degree 1 and γ(2)

L3p;Z×1S1 in degree 2.

βg acts on the generators as

βg(γ(1)

L3p;Zg

× 1S1) = βg(γ(1)

L3p;Zg

)× 1S1 ,

βg(1L3p;Zg

× γ(2)

S1 ) = βg(1L3p;Zg

)× γ(2)

S1 .

The second term vanishes due to (65). The left one can be deduced, using (65),from the following portion of the exact sequence (45)

H 1(L3p, Z)

@g−→ H 1(L3p, Zg)

βg−→ H 2(L3p, Z).

Thus, up to a redefinition of the generator γ(1)

L3p;Zg

,

βg(γ(1)

L3p;Zg

× 1S1) = p

〈p, g〉γ(2)

L3p;Z × 1S1 , βg(1L3

p;Zg× γ

(1)

S1 ) = 0. (67)

Finally, we note that L3p is orientable, hence H 4(L3

p×S1, Z) is torsion-free. In viewof (66), this implies that the cup product is trivial in second degree.

Now let us discuss some special choices for J . For brevity, we write J in theform J = (k1, . . . , kr |m1, . . . , mr).

J = (1|n) ∈ K(n). Here SU(J ) = Zn, the center of SU(n). Moreover, g = n.Variables are ξ ∈ H 1(M, Zn) and α = 1 + α(2), α(2) ∈ H 2(M, Z). The system ofEquations (63) and (64) reads

α(2) = βn(ξ),n(n− 1)

2α(2) = α(2) = c2(P ).

Note that we have used (48) and (49). The first equation yields n α(2) = 0, so thatthe second one requires c2(P ) = 0. Hence, K(P, J ) is nonempty iff P is trivial andis then parametrized by ξ . This coincides with what is known about Zn-subbundlesof SU(n)-bundles.


J = (n|1) ∈ K(n). Here SU(J ) = SU(n), the whole group. Due to g = 1, theonly variable is α = 1 + α(2) + α(4), where α(2j) ∈ H 2j (M, Z), j = 1, 2. Thesystem of Equations (63) and (64) is

α(2) = 0, α(4) = c2(P ).

Thus, K(P, J ) consists of P itself.

J = (1, 1|2, 2) ∈ K(4). One can check that SU(J ) has connected components

{diag(z, z, z−1, z−1) | z ∈ U(1)}, {diag(z, z,−z−1,−z−1) | z ∈ U(1)}.It is, therefore, isomorphic to U(1) × Z2. Variables are ξ ∈ H 1(M, Z2) and αi =1+ α

(2)i , α

(2)i ∈ H 2(M, Z), i = 1, 2. The system of equations under consideration

is

α(2)

1 + α(2)

2 = β2(ξ), α(2)

1 = α(2)

1 + α(2)

2 = α(2)

2 + 4α(2)

1 = α(2)

2 = c2(P ).

Since products including β2(ξ) vanish, by eliminating α(2)2 we obtain

−2α(2)1 = α

(2)1 = c2(P ). (68)

For base manifold M = S4, H 2(M, Z) = 0. Hence, K(P, J ) is nonempty iffc2(P ) = 0, in which case it consists of the (necessarily trivial) U(1) × Z2-bundleover S4.

For M = L3p × S1, in case c2(P ) = 0, K(P, J ) is parametrized by

ξ ∈ H 1(M, Zg) ∼= Z〈2,p〉 ⊕ Z2 and α(2)1 ∈ H 2(M, Z) ∼= Zp.

Otherwise it is empty.For M = S2 × S2, H 1(M, Zg) = 0. Let γ

(2)

S2 be a generator of H 2(S2, Z). Then

H 2(M, Z) ∼= Z⊕ Z is generated by γ(2)

S2 × 1S2 and 1S2 × γ(2)

S2 , whereas H 4(M, Z)

is generated by γ(2)

S2 × γ(2)

S2 . Writing

α(2)

1 = aγ(2)

S2 × 1S2 + b1S2 × γ(2)

S2 (69)

with a, b ∈ Z, Equation (68) becomes

−4abγ(2)

S2 × γ(2)

S2 = c2(P ).

If c2(P ) = 0, there are two series of solutions: a = 0 and b ∈ Z as well as a ∈ Z

and b = 0. Here K(P, J ) is infinite. If c2(P ) = 4lγ(2)

S2 × γ(2)

S2 for some l �= 0, thena = q and b = −l/q, where q runs through the (positive and negative) divisorsof l. Hence, in this case, the cardinality of K(P, J ) is twice the number of divisorsof l. If c2(P ) is not divisible by 4 then K(P, J ) is empty.

Finally, for M = T4 one has

H 1(M, Z2) ∼= Z⊕42 and H 2(M, Z) ∼= Z

⊕6.


Moreover, H 2(M, Z) is generated by elements γ(2)

T4;ij , 1 � i < j � 4, where

γ(2)

T4;12 = γ(1)

S1 × γ(1)

S1 × 1S1 × 1S1, γ(2)

T4;13 = γ(1)

S1 × 1S1 × γ(1)

S1 × 1S1 , etc.,

whereas H 4(M, Z) is generated by γ(4)

T4 = γ(1)

S1 ×γ(1)

S1 ×γ(1)

S1 ×γ(1)

S1 . One can check

γ(2)

T4;ij = γ(2)

T4;kl= εijkl γ

(4)

T4 , (70)

where εijkl denotes the totally antisymmetric tensor in four dimensions. Writing

α(2)1 =

∑1�i<j�4

aij γ(2)

T4;ij (71)

and using (70), Equation (68) yields

−4(a12a34 − a13a24 + a14a23)γ(4)

T4 = c2(P ).

Hence, we find that K(P, J ) is again nonempty iff c2(P ) is divisible by 4, in whichcase it now has always infinitely many elements.

J = (1, 1|2, 3) ∈ K(5). The subgroup SU(J ) of SU(5) consists of matrices ofthe form diag(z1, z1, z2, z2, z2), where z1, z2 ∈ U(1) such that z2

1z32 = 1. We can

parameterize z1 = z3, z2 = z−2, z ∈ U(1). Hence, SU(J ) is isomorphic to U(1).Variables are αi = 1+ α

(2)i , i = 1, 2. The equations to be solved read

2α(2)

1 + 3α(2)

2 = 0, α(2)

1 = α(2)

1 + 3α(2)

2 = α(2)

2 + 6α(2)

1 = α(2)

2 = c2(P ).

Parametrizing α(2)1 = 3η, α

(2)2 = −2η, where η ∈ H 2(M, Z), we obtain

−15η = η = c2(P ).

The discussion of this equation is analogous to that of Equation (68) above.

J = (2, 3|1, 1) ∈ K(5). Here SU(J ) ∼= S(U(2) × U(3)), the symmetry groupof the standard model. In the grand unified SU(5)-model this is the subgroup towhich SU(5) is broken by the heavy Higgs field. Moreover, the subgroup SU(J ) isthe centralizer of the subgroup SU(1, 1|2, 3) discussed above. Variables are αi =1+α

(2)i +α

(4)i , where α

(2j)

i ∈ H 2j (M, Z), i, j = 1, 2. Equations (63) and (64) read

α(2)1 + α

(2)2 = 0, α

(4)1 + α

(4)2 + α

(2)1 = α

(2)2 = c2(P ).

Replacing α(2)

2 = −α(2)

1 we obtain

α(4)2 = c2(P )− α

(4)1 + α

(2)1 = α

(2)1 .

Thus, K(P, J ) can be parametrized by α1 (or α2), i.e., by the Chern class of one ofthe factors U(2) or U(3). Due to the important role S(U(2) × U(3)) is playing insymmetry breaking, this has been known for a long time [20].


J = (2|2). The subgroup SU(J ) of SU(4) consists of matrices D⊕D, D ∈ U(2),(det D)2 = 1. Hence, it has connected components {D ⊕ D | D ∈ SU(2)} and{(iD)⊕ (iD) | D ∈ SU(2)}. One can check that the map SU(2) × Z4 → SU(J ),(D, a) �→ e2πia/4D, induces an isomorphism from (SU(2)× Z4)/Z2 onto SU(J ).Variables are ξ ∈ H 1(M, Z2) and α = 1+ α(2) + α(4). We have

α(2) = β2(ξ), α(2) = α(2) + 2α(4) = c2(P ). (72)

The first equation expresses α(2) in terms of ξ . For example, in case M = L3p × S1,

by expanding ξ = ξLγ(1)

L3p;Zg

× 1S1 + ξS1L3p× γ

(1)

S1 , Equation (67) implies

α(2) = qξLγ(2)

L3p;Z × 1S1 for p = 2q, α(2) = 0 for p = 2q + 1. (73)

The second equation becomes 2α(4) = c2(P ). Thus, K(P, J ) is nonempty iff c2(P )

is even and is then parametrized by ξ ∈ H 1(M, Z2).Let us point out that for M = L3

p × S1, p even, Equation (73) implies thatprincipal SU(J )-bundles which are nontrivial over the factor L3

p have a magneticcharge. This distinguishes SU(J ) from (SU(2)×Z2), because a principal (SU(2)×Z2)-bundle can never have a magnetic charge.

To conclude, we remark that Equations (63) and (64) will be studied system-atically elsewhere. Let us only note the following. Equation (64) always leadsto a diophantine equation. For the base manifolds considered above, the latter isbilinear. For such equations, there exists an algorithm to parameterize the set ofsolutions [33]. The situation is different, for example, for M = CP2. Here theequation obtained from (64) is quadratic and, therefore, substantially harder todiscuss.

6. Holonomy-Induced Bundle Reductions

In the next step of our programme we have to determine which of the Howesubbundles of P are holonomy-induced.

LEMMA 6.1. Let H ⊆ H ′ ⊆ SU(n) be Howe subgroups. If dim H = dim H ′,then H = H ′.

Proof. There exist J, J ′ ∈ K(n) such that H and H ′ are conjugate to SU(J ) andSU(J ′), respectively. Consider U(J ) and U(J ′). Due to H ⊆ H ′, there exists D ∈SU(n) such that D−1U(J )D ⊆ U(J ′). Moreover, by assumption, dim U(J ′) =dim SU(J ′) + 1 = dim SU(J ) + 1 = dim U(J ). Since U(J ′) is connected andD−1U(J )D is closed in U(J ′), equality of dimension implies D−1U(J )D = U(J ′).Then D−1SU(J )D = D−1(U(J )∩ SU(n))D = (D−1U(J )D)∩ SU(n) = U(J ′)∩SU(n) = SU(J ′). It follows H = H ′. ✷THEOREM 6.2. Any Howe subbundle of a principal SU(n)-bundle is holonomy-induced.


Proof. Let P be a principal SU(n)-bundle and let Q be a Howe subbundle of P

with structure group H . Denote the structure group of a connected componentof Q by H . Since H is Howe, H ⊆ C2

SU(n)(H ) ⊆ C2SU(n)(H) = H . Since H and

H have the same dimension, so do C2SU(n)(H ) and H . Then Lemma 6.1 implies

C2SU(n)(H ) = H . Hence the assertion. ✷

For the reader who wonders whether there exist Howe subbundles which arenot holonomy-induced we give an example. Consider the Lie group SO(3). Onechecks that the subgroup H = {13, diag(−1,−1, 1)} is Howe. Thus, the reductionQ = M × H of M × SO(3) is a Howe subbundle. Any connected reduction Q

of Q has the center {13} as its structure group. Since the center is Howe itself,Q · C2

G({13}) = Q �= Q, i.e., Q is not holonomy-induced.

7. Factorization by SU(n)-Action

In Step 4 of our programme to determine Red∗(P ), we actually have to take thedisjoint union of Red(P, H) over all Howe subgroups H of SU(n) and to factorizeby the action of SU(n). Since SU(n)-action on bundle reductions conjugates theirstructure groups, however, it suffices to take the union only over SU(J ), J ∈ K(n):⊔

J∈K(n)

Red(P, SU(J )). (74)

We define

K(P ) =⊔

J∈K(n)

K(P, J ). (75)

We shall denote the elements of K(P ) by L and write them in the form L =(J ;α, ξ), where J ∈ K(n) and (α, ξ) ∈ K(P, J ). Due to Theorem 5.16, the col-lection of characteristic classes {(αJ , ξJ ) | J ∈ K(n)} defines a bijection from (74)onto K(P ). Now we reverse this bijection: For L ∈ K(P ), L = (J ;α, ξ), defineQL ∈ Red(P, SU(J )) by

αJ (QL) = α, ξJ (QL) = ξ. (76)

LEMMA 7.1. Let L, L′ ∈ K(P ), where L = (J ;α, ξ), L′ = (J ′;α′, ξ ′). Thereexists D ∈ SU(n) such that QL′ = QL · D if and only if ξ ′ = ξ and J ′ = σJ ,α′ = σα for some permutation σ .

Proof. We start with some preliminary considerations. For J1, J2 ∈ K(n), weintroduce the notation

N(J1, J2) := {D ∈ SU(n) | D−1SU(J1)D ⊆ SU(J2)}.Any D ∈ N(J1, J2), defines embeddings

hD: U(J1) → U(J2), C �→ D−1CD,

hSD: SU(J1) → SU(J2), C �→ D−1CD.


Let L, L′ be given as in the lemma. Assume that J ′ = σJ for some permutationσ and let D ∈ N(J, J ′). Then hD and hS

D are isomorphisms. One can check that

QL ·D = Q[hS

D]L . Accordingly, the classifying map of QL ·D is

f(QL·D) = BhSD ◦ fQL

, (77)

cf. (19). We decompose hD into a pure permutation of factors and an inner auto-morphism of U(J ) as follows. There exists a permutation σD of 1, . . . , r such thathD maps the σD(i)th factor of U(J ) isomorphically onto the ith factor of U(J ′).Then, in particular,

J ′ = σDJ (78)

(note that σD can differ from σ by a permutation which leaves J invariant). Fur-thermore, there exists C ∈ N(J, J ′) such that prU(J ′)

i ◦ hC = prU(J )

σD(i), ∀i. Then

prU(J ′)i ◦ jJ ′ ◦ hS

C = prU(J )

σD(i) ◦ jJ . (79)

We define B = DC−1. Then BhSD = BhS

B◦BhSC . By construction, B ∈ N(J, J ) and

hB is an automorphism of U(J ) which leaves each factor invariant separately. Onecan check that hB is inner. Then hS

B is an inner automorphism of SU(J ) and can,therefore, be generated by an element of the connected component of the identity.It follows BhS

B = BidSU(J ) ≡ idBSU(J ), hence BhSD = BhS

C, up to homotopy. Thus,(77) becomes

f(QL·D) = BhSC ◦ fQL

. (80)

Now we can compute the characteristic classes of QL ·D in terms of those of QL.Using (80), (39) and (79) we find:

αJ ′(QL ·D) = σDαJ (QL). (81)

In a similar way, using the obvious equality λSJ ′ ◦ hS

C = λSJ , one can check that

ξJ ′(QL ·D) = ξJ (QL). (82)

Now let us turn to the proof of the lemma. First, assume that there exists D ∈ SU(n)

such that QL′ = QL ·D. Then D ∈ N(J, J ′) and J ′ = σJ for some permutation σ .Thus, we can apply the above argumentation. First of all, (78) holds. Moreover, dueto (81),

α′ = αJ ′(QL′) = αJ ′(Q ·D) = σDαJ (QL) = σDα.

Similarly, using (82), one can check that ξ ′ = ξ . This yields the assertion, wherethe desired permutation is σD. Conversely, assume that ξ ′ = ξ and α′ = σα,J ′ = σJ for some permutation σ . Due to Lemma 4.2, there exists D ∈ N(J, J ′).


Obviously, D can be chosen in such a way that σD = σ . Consider QL · D. Dueto (81),

αJ ′(QL ·D) = σαJ (QL) = σα = α′ = αJ ′(QL′).

Similarly, using (82), one can verify ξJ ′(QL · D) = ξJ ′(QL′). Consequently,QL ·D = QL′ . ✷

As suggested by Lemma 7.1, we introduce an equivalence relation on K(P ):Write L ∼ L′ iff ξ ′ = ξ and J ′ = σJ , α′ = σα for some permutation σ . Let K(P )

denote the of equivalence classes.

THEOREM 7.2. The assignment L �→ QL induces a bijection from K(P ) ontoRed∗(P ).

Proof. The assignment L �→ QL induces a surjective map K(P ) → Red∗(P ).Due to Lemma 7.1, it projects to K(P ) and the projected map is injective. ✷

With Theorem 7.2 we have accomplished the determination of Red∗(P ) and,therefore, of the set of orbit types �k. Calculations for the latter can now beperformed entirely on the level of the classifying set K(P ).

8. Example: Gauge Orbit Types for SU(2)

In this section, we are going to determine �k for an SU(2)-gauge theory over thebase manifolds discussed in Subsection 5.10. The set K(2) contains the elements

Ja = (1|2), Jb = (1, 1|1, 1), Jc = (2|1).

Here SU(Ja) = center, SU(Jb) ∼= U(1) (toral subgroup) and SU(Jc) = SU(2). Thestrata of Mk corresponding to the elements of K(P, Ja), K(P, Jb), K(P, Jc) are,in the respective order, those with stabilizers isomorphic to SU(2), U(1), and thegeneric stratum. Accordingly, we shall refer to the first class as SU(2)-strata and tothe second class as U(1)-strata. We have

K(P ) = K(P, Ja) ∪ K(P, Jb) ∪K(P, Jc)

(disjoint union). As we already know, K(P, Ja) is parametrized by ξ ∈ H 1(M, Z2)

if P is trivial and is empty otherwise. Moreover, K(P, Jc) consists of P itself. ForK(P, Jb), variables are αi = 1 + α

(2)i , α

(2)i ∈ H 2(M, Z), i = 1, 2. Equations (63),

(64) read

α(2)1 + α

(2)2 = 0, α

(2)1 = α

(2)2 = c2(P ).

Replacing α(2)

2 we obtain

−α(2)1 = α

(2)1 = c2(P ). (83)


Note that here α(2)1 is just the first Chern class of a reduction of P to the subgroup

U(1). According to this, Equation (83) has been discussed in connection with spon-taneous symmetry breaking of SU(2) to U(1), see [20]. Note also that when passingfrom K(P ) to K(P ), the pairs (α1, α2) and (α2, α1) label the same class of bundlereductions. Hence, solutions α

(2)1 of (83) have to be identified with their negative.

Let us now consider specific choices for the base manifold M.

M = S4. Equation (83) requires c2(P ) = 0. Thus, if P is trivial, K(P ) containsthe Z2-bundle, the U(1)-bundle (both necessarily trivial) and P itself. Accordingly,in Mk there exist, besides the generic stratum, an SU(2)-stratum and a U(1)-stratum. This is well known and was studied in detail, for instance, in [14]. Theauthors found that the two nongeneric strata of Mk can be parametrized by meansof an affine subspace of Ak which is acted upon by the Weyl group of SU(2). Incase P is nontrivial, K(P ) consists of P alone. Accordingly, nongeneric strata donot exist, i.e., Mk is already a manifold.

M = S2 × S2. We use the notation of Subsection 5.10. Writing α(2)1 in the form

(69), Equation (83) becomes

−2abγ(2)

S2 × γ(2)

S2 = c2(P ).

Thus, if P is trivial then K(P ) contains the Z2-bundle, which is trivial, and theU(1)-bundles labelled by a � 0, b = 0 and a = 0, b � 0, i.e., which are trivial overone of the 2-spheres. Accordingly, Mk contains an SU(2)-stratum and infinitelymany U(1)-strata. In case c2(P ) = 2lγ

(2)

S2 × γ(2)

S2 , l �= 0, K(P ) contains the U(1)-bundles with a = q and b = −l/q, where q is a (positive) divisor of m. Hence,here the nongeneric part of Mk consists of finitely many U(1)-strata. In case c2(P )

is odd, Mk consists only of the generic stratum.

M = T4. Writing α(2)1 in the form (71) and using (70), Equation (83) reads

−2(a12a34 − a13a24 + a14a23)γ(4)

T4 = c2(P ). (84)

Hence, the result is similar to that for the case M = S2 × S2. The only differenceis that, due to H 1(M, Z2) ∼= Z

⊕42 , there exist 16 different Z2-bundles which are all

contained in K(P ) for P being trivial. Accordingly, in this case Mk contains 16SU(2)-strata. Moreover, in case c2(P ) = 2lγ

(4)

T4 , l �= 0, the number of solutionsof (84) is infinite. Therefore, in this case there exist infinitely many U(1)-strata.

We remark that pure Yang–Mills theory on T4 has been discussed in [12, 13],where the authors have studied the maximal Abelian gauge for gauge group SU(n).They have found that if P is nontrivial, this gauge fixing is necessarily singularon Dirac strings joining magnetically charged defects in the base manifold. It isan interesting question whether there is a relation between such defects and thenongeneric strata of Mk.


M = L3p×S1. Here (83) requires c2(P ) = 0. Thus, if P is trivial, K(P ) contains

the Z2-bundles which are labelled by the elements of H 1(L3p × S1, Z2), i.e., by

Z2 ⊕ Z2 in case p �= 0 and even or by Z2 otherwise, as well as the U(1)-bundles,which are labelled by α

(2)1 ∈ H 2(L3

p × S1, Z) ∼= Zp, modulo α(2)1 ∼ −α

(2)1 . Ac-

cordingly, if p is even, the nongeneric part of Mk consists of four SU(2)-strata and(p/2+ 1) U(1)-strata. If p is odd, it consists of two SU(2)-strata and ((p + 1)/2)

U(1)-strata. If P is nontrivial, nongeneric strata do not exist.

9. Application: Kinematical Nodes in Yang–Mills Theory withChern–Simons Term

Following [3], we consider Yang–Mills–Chern–Simons theory with gauge groupSU(n) in the Hamiltonian approach. The Hamiltonian in Schrödinger representa-tion is given by

H = −J

2

∫M

d2x√h

Tr

∣∣∣∣ δ

δAµ

+ iK

4πεµνAν

∣∣∣∣2 + 1

4J

∫M

d2x√

hTr(FµνFµν).

Geometrically, Aµ and Fµν are the local representatives of a connection A andits curvature F in a (necessarily trivial) SU(n)-bundle P over the two-dimensionalspace M. Physical states are given by functionals ψ on the space Ak of connectionsin P which obey the Gauss law

∇µ

A

δ

δAµ(x)ψ(A) = iK

4πεµν∂µAν(x)ψ(A). (85)

Here ∇A denotes the covariant derivative w.r.t. A. In [3] it was shown that if A

carries a nontrivial magnetic charge, i.e., if it can be reduced to some reductionof P with nontrivial first Chern class, all physical states obey ψ(A) = 0. Such aconnection is called a kinematical node. (Note that there exist also dynamical nodeswhich differ from state to state.) The authors of [3] argue that nodal gauge fieldconfigurations are relevant for the confinement mechanism. In the following, weshall show that being a node is a property of strata. For that purpose, we reformulatethe result of [3] in our language.

THEOREM 9.1. Let A ∈ Ak have orbit type [L] ∈ K(P ), where L = (J ;α, ξ).If α

(2)i �= 0 for some i then A is a kinematical node, i.e., ψ(A) = 0 for all physical

states ψ .Proof. The proof follows the lines of [3]. By assumption, A can be reduced to

a connection on QL ∈ Red(P, SU(J )). M being a compact orientable 2-manifold,H 2(M, Z) ∼= Z. Hence, α

(2)i = ciγ

(2), where ci ∈ Z and γ (2) is a generator ofH 2(M, Z). We define a 1-parameter subgroup {Ot | t ∈ R} of SU(n) by

Ot =(

exp

{ic1

k1t

}1k1 ⊗ 1m1

)⊕ · · · ⊕

(exp

{icr

kr

t

}1kr⊗ 1mr

).


Due to (63),

(m1c1 + · · · +mrcr)γ(2) = g(m1α

(2)1 + · · · + mrα

(2)r ) = g βg(ξ) = 0.

Hence,

det Ot = exp{it (m1c1 + · · · +mrcr)} = 1,

so that Ot ⊆ SU(n), ∀t . Commuting with SU(J ) for all t , Ot defines a 1-parametersubgroup {Ot | t ∈ R} of Gk+1 by

Ot(q) = Ot , ∀q ∈ QL, t ∈ R.

Each element of this subgroup is constant on Q. Hence, so is the generator φ =O(0):

φ(q) = i

[(c1

k11k1 ⊗ 1m1

)⊕ · · · ⊕

(cr

kr

1kr⊗ 1mr

)], ∀q ∈ QL. (86)

In particular,

∇Aφ = 0. (87)

According to this, for any state ψ ,∫M

Tr

(φ∇µ

A

δ

δAµ

)ψ(A) = −

∫M

Tr

((∇µ

Aφ)δ

δAµ

)ψ(A) = 0.

For physical states, the Gauss law implies∫M

Tr(φ dA)ψ(A) = 0. (88)

Using (87), as well as the structure equation F = dA+ 12 [A, A], we obtain∫

M

Tr(φ dA) = 2∫

M

Tr(φF). (89)

Since A is reducible to QL, F has block structure (F1 ⊗ 1m1)⊕ · · · ⊕ (Fr ⊗ 1mr)

with Fj being (kj × kj )-matrices. Thus, using (86),∫M

Tr(φF) = ir∑

j=1

mj

kj

cj

∫M

Tr Fj . (90)

Now cj being just the first Chern classes of the elementary factors of the U(k1)×· · · × U(kr)-bundle Q

[U(J )]L , we have∫

M

Tr Fj = −2π icj , j = 1, . . . , r.


Inserting this into (90) and the latter into (89) we obtain∫M

Tr(φ dA) = 4π

r∑j=1

mj

kj

(cj )2.

Thus, in view of (88),

r∑j=1

mj

kj

(cj )2ψ(A) = 0. (91)

It follows that if one of the cj is nonzero then ψ(A) = 0 for all physical states ψ ,i.e., A is a kinematical node. This proves the theorem. ✷

Remark. Let us compare (91) with Formula (6) in [3]. Define k′i = kimi andm′

i = 1. Then J ′ = (k′, m′) ∈ K(n) and U(J ) ⊆ U(J ′). Consider the extension

Q[U(J ′)]L ⊆ P . One can check that the elementary factors of this bundle have first

Chern classes c′i = mici . Inserting k′i , l′i , and c′i into (91) one obtains formula (6)

in [3]. In fact, the authors of [3] use that A is reducible to Q[U(J ′)]L , rather than that it

is even reducible to QL. This argument being ‘coarser’ than ours, it still suffices toprove that any connection which is reducible to a bundle reduction with nontrivialmagnetic charge is a kinematical node.

As a consequence of Theorem 9.1, one can speak of nodal and nonnodal strata.This information can be read off directly from the labels of the strata. Let us discussthis in some more detail. Let J ∈ K(n) be given and consider Equation (63) (Equa-tion (64) is trivially satisfied). Variables are ξ ∈ H 1(M, Zg) and α

(2)i ∈ H 2(M, Z),

i = 1, . . . , r. Since H 2(M, Z) is torsion-free, βg is trivial. Thus, the equation to besolved is

E(2)m (α) = 0. (92)

Writing α(2)i = ciγ

(2) again, (92) becomes

r∑i=1

mici = 0.

The set of solutions is a subgroup Gm ⊆ Z⊕r . According to Theorem 9.1, the

nonnodal strata are parametrized by ξ and the neutral element of Gm, whereas thenodal strata are labelled by ξ and all the other elements of Gm. For example, in thecase of SU(2) we obtain the following. For J = (1|2), Gm = {0} ⊆ Z, hence allstrata are nonnodal. For J = (1, 1|1, 1), Gm = {(c,−c) | c ∈ Z} ⊆ Z

⊕2. Sincealso ξ = 0, each value of c labels one stratum. That corresponding to c = 0 isnonnodal, the others are nodal. For J = (2|1), we have the generic stratum, whichis nonnodal.


10. Summary

Starting from a principal SU(n)-bundle P over a compact connected orientableRiemannian 4-manifold M, we have derived a classification of the orbit types ofthe action of the group of gauge transformations of P on the space of connectionsin P . Orbit types are known to label the elements of the natural stratification, givenby Kondracki and Rogulski [23], of the gauge orbit space associated to P . Theinterest in this stratification is due to the fact that the role of nongeneric strata ingauge physics is not clarified yet.

In order to accomplish the classification, we have utilized that orbit types are inone-to-one correspondence with a certain class of bundle reductions of P (calledholonomy-induced), factorized by isomorphy and the natural action of the struc-ture group. We have shown that such classes of bundle reductions are labelled bysymbols [(J ;α, ξ)], where J = ((k1, . . . , kr ), (m1, . . . , mr)) is a pair of sequencesof positive integers obeying

r∑i=1

kimi = n,

α = (α1, . . . , αr), where αi ∈ H even(M, Z) are admissible values of the Chernclass of U(ki)-bundles over M, and ξ ∈ H 1(M, Zg) with g being the greatestcommon divisor of (m1, . . . , mr). The cohomology elements αi and ξ are subjectto the relations

r∑i=1

mi

gα

(2)i = βg(ξ),

αm11 = . . . = αmr

r = c(P ),

where βg: H 1(M, Zg) → H 2(M, Z) is the connecting (i.e., Bockstein) homomor-phism associated to the short exact sequence of coefficient groups

0 → Z → Z → Zg → 0.

Furthermore, for any permutation σ of {1, . . . , r}, the symbols [(J ;α, ξ)] and[(σJ ;σα, ξ)] have to be identified.

The result obtained enables one to determine which strata are present in thegauge orbit space, depending on the topology of the base manifold and the topolog-ical sector, i.e., the isomorphism class of P . For some examples we have discussedthis dependence in detail. We have also shown that our result can be used to refor-mulate a sufficient condition, on a connection to be a node for all physical states inYang–Mills Theory with Chern–Simons term, derived in [3].

Our result may be viewed as one more step towards a systematic investigationof the physical effects related to nongeneric strata of the gauge orbit space.

We remark that orbit types still carry more information about the stratifica-tion structure. Namely, their partial ordering encodes how the strata are patched


together in order to build up the gauge orbit space (cf. [23, Thm. (4.3.5)]). A deriva-tion of this partial ordering can be found in [37].

Appendix: The Eilenberg–MacLane Spaces K(Z, 2) and K(Zg, 1)

In this appendix, we construct a model for each of the Eilenberg–MacLane spacesK(Z, 2) and K(Zg, 1) and derive the integer-valued cohomology of these spaces.Consider the natural free action of U(1) on the sphere S∞ which is induced fromthe natural action of U(1) on S2n−1 ⊂ C

n. The orbit space of this action is thecomplex projective space CP∞. Moreover, by viewing Zg as a subgroup of U(1),this action gives rise to a natural free action of Zg on S∞. The orbit space of thelatter is the lens space L∞g . By construction, one has principal bundles

U(1) ↪→ S∞ −→ CP∞, (93)

Zg ↪→ S∞ −→ L∞g . (94)

Due to πi(S∞) = 0, ∀i, the exact homotopy sequences induced by (93), (94) yield

πi(CP∞) = πi−1(U(1)) ={

Z, i = 2,

0, i �= 2,

πi(L∞g ) = πi−1(Zg) =

{Zg, i = 1,

0, i = 2, 3, . . . ,

respectively. As a consequence, CP∞ is a model of K(Z, 2) and L∞g is a model ofK(Zg, 1). In particular,

H i(K(Z, 2), Z) = H i(CP∞, Z) ={

Z, i even,

0, i odd,(95)

see [8, Ch. VI, Prop. 10.2], and

H i(K(Zg, 1), Z) = H i(L∞g , Z) = Z, i = 0,

Zg, i �= 0, even,

0, i �= 0, odd,

(96)

see [35, §II.7.7] (and use the Universal Coefficient Theorem). We notice that thevanishing of all homotopy groups of S∞ also implies that the bundles (93) and (94)are universal for U(1) and Zg, respectively. Hence, CP∞ and L∞g are models ofBU(1) and BZg, respectively. For BZg, this has been used in the proof ofLemma 5.9.

Acknowledgements

The authors would like to thank S. Boller, C. Fleischhack, S. Kolb, and A. Stroh-maier for interesting discussions. They are also grateful to T. Friedrich, J. Hilgert,


and H.-B. Rademacher for useful suggestions, as well as to L. M. Woodward,C. Isham, and M. Cadek who have been very helpful in providing specific infor-mation.

References

1. Abbati, M. C., Cirelli, R., Manià, A. and Michor, P.: The Lie group of automorphisms of aprincipal bundle, J. Geom. Phys. 6(2) (1989), 215–235.

2. Abbati, M. C., Cirelli, R. and Manià, A.: The orbit space of the action of gauge transformationgroup on connections, J. Geom. Phys. 6(4) (1989), 537–557.

3. Asorey, M., Falceto, F., López, J. L. and Luzón, G.: Nodes, monopoles, and confinement in2+ 1-dimensional gauge theories, Phys. Lett. B 345 (1995), 125–130.

4. Asorey, M.: Maximal non-Abelian gauges and topology of the gauge orbit space, NuclearPhys. B 551 (1999), 399–424.

5. Avis, S. J. and Isham, C. J.: Quantum field theory and fibre bundles in a general space-time,In: M. Levy and S. Deser (eds), Recent Developments in Gravitation (Cargèse 1978), PlenumPress, New York, 1979, pp. 347–401.

6. Borel, A.: Topics in the Homology Theory of Fibre Bundles, Lecture Notes in Math. 36,Springer, New York, 1967.

7. Bredon, G. E.: Introduction to Compact Transformation Groups, Academic Press, New York,1972.

8. Bredon, G. E.: Topology and Geometry, Springer, New York, 1993.9. Dodson, C. T. J. and Parker, P. E.: A User’s Guide to Algebraic Topology, Kluwer Acad. Publ.,

Dordrecht, 1997.10. Emmrich, C. and Römer, H.: Orbifolds as configuration spaces of systems with gauge

symmetries, Comm. Math. Phys. 129 (1990), 69–94.11. Fomenko, A. T., Fuchs, D. B. and Gutenmacher, V. L.: Homotopic Topology, Akadémiai Kiadó,

Budapest, 1986.12. Ford, C., Tok, T. and Wipf, A.: Abelian projection on the torus for general gauge groups,

Nuclear Phys. B 548 (1999), 585–612.13. Ford, C., Tok, T. and Wipf, A.: SU(N)-gauge theories in Polyakov gauge on the torus,

Phys. Lett. B 456 (1999), 155–161.14. Fuchs, J., Schmidt, M. G. and Schweigert, C.: On the configuration space of gauge theories,

Nuclear Phys. B 426 (1994), 107–128.15. Heil, A., Kersch, A., Papadopoulos, N. A., Reifenhäuser, B. and Scheck, F.: Structure of the

space of reducible connections for Yang–Mills theories, J. Geom. Phys. 7(4) (1990), 489–505.16. Heil, A., Kersch, A., Papadopoulos, N. A., Reifenhäuser, B. and Scheck, F.: Anomalies from

nonfree action of the gauge group, Ann. Phys. 200 (1990), 206–215.17. Hirzebruch, F.: Topological Methods in Algebraic Geometry, Springer, New York, 1978.18. Howe, R.: θ-series and invariant theory, In: Automorphic Forms, Representations, and L-

functions, Proc. Sympos. Pure Math. 33, part 1, Amer. Math. Soc., Providence, 1979,pp. 275–285.

19. Husemoller, D.: Fibre Bundles, McGraw-Hill, New York, 1966; Springer, New York, 1994.20. Isham, C. J.: Space-time topology and spontaneous symmetry breaking, J. Phys. A 14 (1981),

2943–2956.21. Kobayashi, S. and Nomizu, K.: Foundations of Differential Geometry, Vol. I, Wiley-

Interscience, New York, 1963.22. Kondracki, W. and Rogulski, J.: On the notion of stratification, Demonstratio Math. 19 (1986),

229–236.


23. Kondracki, W. and Rogulski, J.: On the stratification of the orbit space for the action of au-tomorphisms on connections, Dissertationes Math. 250, Panstwowe Wydawnictwo Naukowe,Warsaw, 1986.

24. Kondracki, W. and Sadowski, P.: Geometric structure on the orbit space of gauge connections,J. Geom. Phys. 3(3) (1986), 421–433.

25. Massey, W. S.: A Basic Course in Algebraic Topology, Springer, New York, 1991.26. Mitter, P. K. and Viallet, C.-M.: On the bundle of connections and the gauge orbit manifold in

Yang–Mills theory, Comm. Math. Phys. 79 (1981), 457–472.27. Moeglin, C., Vignéras, M.-F. and Waldspurger, J.-L.: Correspondances de Howe sur un corps

p-adique, Lecture Notes in Math. 1291, Springer, New York, 1987.28. Palais, R. S.: Foundations of Global Nonlinear Analysis, Benjamin, New York, 1968.29. Przebinda, T.: On Howe’s duality theorem, J. Funct. Anal. 81 (1988), 160–183.30. Rubenthaler, H.: Les paires duales dans les algèbres de Lie réductives, Astérisque 219 (1994).31. Schmidt, M.: Classification and partial ordering of reductive Howe dual pairs of classical Lie

groups, J. Geom. Phys. 29 (1999), 283–318.32. Singer, I. M.: Some remarks on the Gribov ambiguity, Comm. Math. Phys. 60 (1978), 7–12.33. Skolem, T.: Diophantische Gleichungen, Springer, Berlin, 1938.34. Steenrod, N.: The Topology of Fibre Bundles, Princeton Univ. Press, Princeton, NJ, 1951.35. Whitehead, G. W.: Elements of Homotopy Theory, Grad. Texts in Math. 61, Springer, New

York, 1978.36. Woodward, L. M.: The classification of principal PU(n)-bundles over a 4-complex, J. London

Math. Soc. (2) 25 (1982), 513–524.37. Rudolph, G., Schmidt, M. and Volobuev, I. P.: Partial ordering of gauge orbit types for SU(n)-

gauge theories, J. Geom. Phys. 42 (2002), 106–138.


243

On the Essential Spectrum of a Class of SingularMatrix Differential Operators. I: QuasiregularityConditions and Essential Self-adjointness

PAVEL KURASOV1 and SERGUEI NABOKO2

1Dept. of Mathematics, Lund Institute of Technology, Box 118, 221 00 Lund, Sweden.e-mail: [email protected]. of Mathematical Physics, St. Petersburg Univ., 198904 St. Petersburg, Russia.e-mail: [email protected]

(Received: 5 December 2001; in final form: 26 August 2002)

Abstract. The essential spectrum of singular matrix differential operator determined by the operatormatrix

− ddx ρ(x)

ddx + q(x) d

dxβx

−βx

ddx

m(x)

x2

is studied. It is proven that the essential spectrum of any self-adjoint operator associated with thisexpression consists of two branches. One of these branches (called regularity spectrum) can beobtained by approximating the operator by regular operators (with coefficients which are boundednear the origin), the second branch (called singularity spectrum) appears due to singularity of thecoefficients.

Mathematics Subject Classifications (2000): Primary: 47A10, 76W05; secondary: 34L05, 47B25.

Key words: essential spectrum, quasiregularity conditions, Hain–Lüst operator.

1. Introduction

Systems of ordinary and partial differential and pseudodifferential equations is asubject of interest for many mathematicians (see [19] and numerous referencestherein). Matrix ordinary differential operators of mixed order appear in manyproblems of theoretical physics: hydrodynamics, plasma physics, quantum fieldtheory, and others. Mathematically rigorous treatment of such problems has beencarried out by several authors: J. A. Adam, V. Adamyan, J. Descluox, G. Gey-monat, G. Grubb, T. Kako, H. Langer, A. E. Lifchitz, R. Mennicken, M. Möller,G. D. Raikov, A. Shkalikov, and others [1, 2, 4, 5, 8–11, 15, 17, 20–23, 28, 29, 31,37]. Matrix differential operators with singular coefficients are of special interestin plasma physics, for example so-called force operators describing equilibriumstate of plasma in toroidal region are exactly of this kind [20]. A more general

244 PAVEL KURASOV AND SERGUEI NABOKO

class of 3 × 3 matrix differential operators with singularities was considered byV. Hardt, R. Mennicken, and S. Naboko [17], where a new branch of the essentialspectrum determined by the singularity was observed and described. This newbranch had been predicted by J. Descloux and G. Geymonat [5]. To study theessential spectrum of the operator, so-called quasiregularity conditions were intro-duced ([17]). These conditions are necessary and sufficient for the boundednessof the essential spectrum of the singular operator. A different approach to thisclass of matrix operators satisfying the quasiregularity conditions was developedby M. Faierman, R. Mennicken, and M. Möller [10]. Recently, R. Mennicken,S. Naboko, and Ch. Tretter suggested clarifying approach to study this class ofsingular operators ([30]). It was discovered that the new branch of the essentialspectrum can be characterized as the zero set for the symbol of the asymptoticHain–Lüst operator introduced in [30]. It should be mentioned that in the newapproach, the authors used Proposition A.1 from the current paper.

Investigation of the essential spectrum of differential and partial differential op-erators attracts attention of many scientists ([40, 41, 44]). For example the spectrumof pseudodifferential operators with piecewise continuous symbols has been inves-tigated by S. C. Power [35, 36]. In [14] (Chapter 3), it is shown how to calculatethe essential spectrum for pseudodifferential boundary value problems using theprincipal interior and boundary symbol operators.

A new class of matrix differential operators with singular coefficients is intro-duced and investigated in this paper. This class consists of 2×2 matrices instead ofthe 3 × 3 operator matrices studied in [30], which is a formal simplification. (Themethod elaborated in the paper can be applied to m × m operator matrices.) Butall essential features of the problem are still present. Additionally the singularitiesof the matrix elements are distributed in a different way. We decided to study thisclass of singular operators in order to illustrate the mechanism of the appearanceof the additional branch of essential spectrum using the most explicit example.This helps us to avoid tedious calculations and at the same time preserves themain features of the original problem. For this reason we tried to develop a properCalkin calculus (see Appendix B), which allows one to justify calculations from[17, 20–22] being incomplete. On the other hand, employment of Calkin calculusmakes all calculations transparent and easier. For example, the authors of [4], in-vestigating nonsingular operator matrices, used subtle results from operator theorydue to P. E. Sobolevskii [26]. Developing these methods, some new results onBanach space operators were obtained. These results on the spectrum of the sumof three operators are of an abstract nature and can be used in other problems aswell. Investigating this problem, we tried to elaborate a new general approach tosingular matrix differential operators. We hope to be able to apply this methodto most general singular matrix differential operators including partial differentialoperators.

The operator under investigation is determined formally by the following ex-pression:

ON THE ESSENTIAL SPECTRUM OF MATRIX DIFFERENTIAL OPERATORS I 245

− d

dxρ(x)

d

dx+ q(x) d

dx

β

x

−βx

d

dx

m(x)

x2

. (1)

We use this form of the matrix differential operator in order to display explicitlythe singularities of three matrix elements at the origin. The most interesting (andcomplicated) case is when the functions β and m do not vanish at the origin.Therefore, the operator defined by the functions β andm having zeroes at the originof order 1 and 2 respectively, will be called regular. In this case, all singularitiesare artificial. The essential spectrum of the corresponding operator can easily beinvestigated using the methods of [4]. All other operators from the described classwill be called singular and we are going to concentrate our attention on the caseof singular operators only. It is clear that the matrix symbol does not determine theself-adjoint operator uniquely even in the regular case. The extension theory, of theminimal operator in the regular case has been developed by H. de Snoo [43] and inthe case of nonsingular leading matrix coefficient in [38, 39].

Our interest in singular problem is motivated by the new spectral phenomenonwhich can be observed in this case: the essential spectrum of any selfadjoint op-erator corresponding to the symbol (1) in L2[0, 1] ⊕ L2[0, 1] cannot be describedas a limit of the essential spectra of the operators determined by the same symbolin L2[ε, 1] ⊕ L2[ε, 1] as ε → +0. Such limit determines only a certain part ofthe essential spectrum of the operator in L2[0, 1] ⊕ L2[0, 1]. An additional branchof the essential spectrum appears due to the singularity of the coefficients at theorigin. Trivial counterpart of this phenomena is well known for infinite intervals,since for example the essential spectrum of −(d2/dx2) on a finite interval [−an, bn]is empty and therefore does not give the essential spectrum of −(d2/dx2) on thewhole line when an, bn → ∞. The phenomenon described in the current note ismore sophisticated and is due to rather complicated interplay between the com-ponents of the matrix differential operator. On the other hand, the coefficient ofthe matrix determining the operator have singularities at the boundary points. Thisnew branch of essential spectrum is absent in the case of regular operators, sincethe limit procedure for the essential spectrum described above gives the correctanswer in the regular case. Spectral analysis in the regular case is well knownand can be carried out using methods developed in [4, 15]. In what follows, thetwo branches of the essential spectrum will be called the regularity spectrum andsingularity spectrum, respectively. We introduce quasiregularity conditions for thesingular operator which guarantees boundedness of the regularity spectrum. Thequasiregularity conditions determine a special class of singular matrix differen-tial operators for which we are able to calculate the essential spectra. Note thatin many physical applications, i.e. in plasma physics ([20]), these conditions arefulfilled.

The singularities of the operator coefficients at the origin play an important roleeven at the stage of the definition of the self-adjoint operator corresponding to the


formal expression (1). The indices of the minimal differential operator producedby the singular point are investigated by considering the extension of the minimaloperator to the set of functions satisfying certain symmetric boundary condition atthe regular point. (In this way the singular x = 0 and regular x = 1 endpointsare treated separately and in different ways.) It is proven that this extended op-erator has trivial deficiency indices (is essentially self-adjoint) if and only if thequasiregularity conditions are satisfied and β(0) = 0. (The condition β(0) = 0together with the quasiregularity condition (8) imply for smooth coefficients thatm(0) = m′(0) = 0 and therefore that the operator L is not singular.) If at leastone of the quasiregularity conditions is not satisfied or the function β vanishesat the origin then the deficiency indices of the described extended operator arenontrivial like it is in the regular case. We would like to note that the quasiregularityconditions introduced originally to guarantee boundedness of the regular branch ofthe essential spectrum play an important role in the investigation of the deficiencyindices. (Note that the name quasiregularity conditions has nothing to do withthe regularity of the extension problem for the operator. It refers to the essentialspectrum only.)

After the family of self-adjoint operators corresponding to the formal expres-sion (1) is determined, we discuss the transformation of the operator using theexponential map of the interval [0, 1] onto the half-infinite interval [0,∞). Thismap transforms the singular point at the origin to a point at ∞ and enables us touse the standard Fourier transform in L2(R). So the reason to use this exponentialmap is pure technical.

Since we are interested in the essential spectrum of the corresponding self-adjoint operators, the choice of the boundary conditions in the limit circle caseis not important. The difference between the resolvents of any two operators fromthis family is a finite rank operator. To calculate the essential spectrum of anysuch selfadjoint operator we use the even stronger fact that the essential spectra ofany two self-adjoint operators coincide if the difference between their resolvents iscompact (Weyl theorem). We develop a so-called cleaning procedure which enablesone to reduce the calculation of the essential spectrum of the complicated matrixdifferential operator given by (1) to the calculation of the essential spectrum of acertain asymptotic singular operator with real coefficients. The singular coefficientsof the asymptotic operator are chosen to have the same singularities as those of theoriginal operator. In other words the asymptotic operator is chosen so that the dif-ference between the resolvents of the original and asymptotic operators is compact.The Hain–Lüst operator can be considered as a regularized determinant of the 2×2matrix differential operator (1), and it plays a very important role in the cleaningprocedure. In the considered case the Hain–Lüst operator is an ordinary (scalar)second order differential operator in L2(R+). We hope that the approach developedin the current paper can be applied to more general operators including arbitrarydimension matrix differential operators and matrix partial differential operators.The method of cleaning of the resolvent modulo compact operators is of general


nature. Several abstract lemmas proven in the present paper can be applied withouteven minor changes.

To calculate the essential spectrum of the asymptotic operator we use the factthat its resolvent is equal to the separable sum of two pseudodifferential operators.We call the sum of two pseudodifferential operators separable if the symbol ofone of these two operators depends only on the space variable, and the symbolof the other operator depends only on the momentum variable. Calculation of theessential spectrum of such operators is based on Proposition A.1 from Appendix A.

We observe that the essential spectrum of the model operator under consider-ation coincides with the set of zeroes of the symbol of the asymptotic Hain–Lüstoperator. That operator is a modified version of the original Hain–Lüst operatorwhich preserves information on the behavior of the coefficients at the singular pointonly. This operator has a more simple expression: it is a second-order differentialoperator with constant coefficients. Unfortunately all information concerning theregularity spectrum disappears during this rectification. This probably general re-lation between the symbol of the asymptotic Hain–Lüst operator and the singularityspectrum will be investigated in one of the forthcoming publications.

The methods developed in this article can easily be extended to include differen-tial operators determined by operator matrices of higher dimension. For example,the case when the coefficient m appearing in (1) is a matrix can easily be investi-gated. The developed methods can help to study matrix partial differential operatorsas well. These subjects will be discussed in a future publication.

2. The Minimal Operator

Let us consider the linear operator defined by the following operator valued 2 × 2matrix

L :=

− d

dxρ(x)

d

dx+ q(x) d

dx

β

x

−βx

d

dx

m(x)

x2

, (2)

where the real-valued functions ρ(x), q(x), β(x), and m(x) are continuously dif-ferentiable in the closed interval [0, 1]

ρ, q, β,m ∈ C2[0, 1]. (3)

In addition we suppose that the density function ρ is positive (definite)

ρ(x) � ρ0 > 0. (4)

Certainly these conditions on the coefficients are far from being necessary for ouranalysis, but we assume these conditions in order to avoid unnecessary complica-tions. In this way we are able to present certain new ideas explicitly without gettingthe most optimal result.


The operator matrix (2) determines rather complicated matrix differential op-erator. Indeed in its formal determinant which controls the spectrum of the wholeoperator the differential order of the formal product of the diagonal elements(

− d

dxρ(x)

d

dx+ q(x)

)m(x)

x2

coincides with that of the formal product of the antidiagonal elements

d

dx

β

x

(−βx

d

dx

).

The same holds true for the orders of the singularities at the origin. These relationscan be expressed by the diagrams 2 + 0 = 1 + 1 for the order of differentialoperators and 0 + 2 = 1 + 1 for the orders of the power-like singularities at theorigin. These conditions imply that the nondiagonal coupling cannot be consideredas a weak perturbation of the diagonal part of the operator and therefore no existingperturbation theory can be applied to the study of the operator. The aim of this arti-cle is to describe new spectral phenomena appearing due to this interplay betweenthe singularities.

The operator matrix given by (2) does not determine unique self-adjoint op-erator in the Hilbert space H = L2[0, 1] ⊕ L2[0, 1]. To describe the family ofself-adjoint operators corresponding to (2) let us consider the minimal operatorLmin with the domain C∞

0 (0, 1)⊕C∞0 (0, 1). The operator Lmin is symmetric but is

not self-adjoint. Let us keep the same notation for the closure of the operator.Any self-adjoint operator corresponding to the operator matrix (2) is an exten-

sion of the minimal operator Lmin. It will be shown in Section 4 that the deficiencyindices of Lmin are finite and all self-adjoint extensions of the operator can bedescribed by certain boundary conditions at the end points of the interval [0, 1].In what follows we are going to consider local boundary conditions only. Suchboundary conditions do not connect the boundary values of functions at differentend point of the interval. As usual each self-adjoint extension of the operator Lmin

is a restriction of the adjoint operator L∗min ≡ Lmax, which is defined by the same

operator matrix (2) on the domain of functions from W 22 [0, 1] ⊕ W 1

2 [0, 1] ⊂ Hsatisfying the following two additional conditions ([33])

− d

dxρ(x)

d

dxu1 + qu1 + d

dx

β

xu2 ∈ L2[0, 1];

−βx

d

dxu1 + m

x2u2 ∈ L2[0, 1].

Since the original operator Lmin has finite deficiency indices, the difference be-tween the resolvents of any two self-adjoint extensions of Lmin is a finite rankoperator. Therefore all these self-adjoint operators have just the same essentialspectrum by the Weyl theorem [24].


3. Quasiregularity Conditions

Consider an arbitrary self-adjoint extension L of the operator Lmin. The essentialspectrum of the operator L will be denoted by σess(L) in what follows. One part ofσess(L) can be calculated using the Glazman splitting method (see [3]) already atthis stage. Indeed consider the operator L0(ε) being the restriction of the operatorL to the domain

Dom(L0(ε)) ={F = (f1, f2) ∈ Dom(L) : f1(ε) = d

dxf1(ε) = f2(ε) = 0

}.

Consider the following decomposition of the Hilbert space

L2[0, 1] = L2[0, ε] ⊕ L2[ε, 1].The corresponding decomposition of the Hilbert space H is defined as follows

H = Hε ⊕ H ε = (L2[0, ε] ⊕ L2[0, ε])⊕ (L2[ε, 1] ⊕ L2[ε, 1]).Using this decomposition the operator L0(ε) can be represented as an orthogonalsum of two symmetric operators acting in Hε and H ε respectively. The point x =ε is regular for the operator matrix (2) and one of the self-adjoint extensions ofthe operator L0(ε) is defined by Dirichlet boundary conditions at x = ε±. (Thefact that the Dirichlet boundary condition at any regular point determines a self-adjoint extension is not trivial for matrix differential operators and has been provenrigorously in [43].) Let us denote this extension by L(ε).

The difference between the resolvents of the operators L(ε) and L is at most arank 2 operator. Therefore the essential spectra of these two operators coincide. Inparticular the essential spectrum of the operator L contains the essential spectrumof the operator L(ε) restricted to the subspace H ε = L2[ε, 1] ⊕ L2[ε, 1]

σess(L) ⊃ σess(L(ε)|Hε ), ε ∈ (0, 1). (5)

The restricted operator L(ε)|Hε is a regular matrix self-adjoint operator and itsessential spectrum can be calculated using the results of [4] (Theorem 4.5)

σess(L(ε)|L2(ε,1)) = Rangex∈[ε,1]

(m(x)

x2− β(x)2

x2ρ(x)

). (6)

For any ε > 0 the essential spectrum of L(ε)|Hε fills in a certain finite interval,since the functions m,β, and ρ−1 are finite and therefore bounded on [ε, 1]. Sinceobviously

σess(L) ⊃⋃ε>0

σess(L(ε)|Hε ) = Rangex∈(0,1]

(m(x)

x2− β(x)2

x2ρ(x)

), (7)

the essential spectrum of L is bounded only if the following quasiregularity condi-tions hold

ρm− β2|x=0 = 0,d

dx(ρm− β2)|x=0 = 0. (8)


The quasiregularity conditions appeared first in [17] and were also used later in[9, 10]. Note that the function (ρm− β2)/x2 is related to the leading coefficient ofthe formal determinant of the matrix L (2).

The role of the quasiregularity conditions is explained by the following state-ment based on formula (51) to be proven in Section 8.

LEMMA 3.1. Under the assumptions (3) and (4) on the coefficients ρ, β,m, andq the quasiregularity conditions are fulfilled if and only if the essential spectrum ofat least one (and, hence, any) self-adjoint extension of Lmin is bounded.

Proof. Formula (7) implies that quasiregularity conditions are fulfilled if theessential spectrum for at least one self-adjoint extension of Lmin. Here we used thatthe coefficients satisfy (3). On the other hand, formula (51) valid for any operatormatrix satisfying the quasiregularity conditions implies the boundedness of theessential spectrum for all self-adjoint extensions of Lmin. The lemma is proven,provided formula (51) holds true. ✷

In what follows we are going to call the matrix L quasiregular if the quasi-regularity conditions (8) on the coefficients are satisfied. Regular matrices form asubset of quasiregular operator matrices. The subfamily of regular matrices can becharacterized by one of the following two additional conditions

m(0) = 0 ∨ β(0) = 0. (9)

Really each of these conditions together with the first quasiregularity condition im-ply the other one. Then the second quasiregularity condition implies thatm′(0) = 0. Hence, the corresponding matrix is regular, since m(0) = m′(0) =β(0) = 0. Therefore we are going to concentrate our attention on the case ofquasiregular matrices which are not regular, since the regular matrices have beenstudied earlier ([43]).

4. Deficiency Indices

Self-adjoint extensions of the minimal operator Lmin are investigated in this section.These extensions can be described by certain (generalized) boundary conditions onthe functions from the domain of the extended operator. These boundary conditionsrelates the boundary values at the endpoints x = 0 and x = 1. We restrict ourstudies to local boundary conditions. The boundary conditions are called local ifthey do not join together the boundary values at different points.

Every self-adjoint extension of the operator Lmin is a certain restriction of theadjoint operator L∗

min. To calculate the adjoint operator it is enough to considerthe operator Lmin restricted to the set of functions from C∞

0 (0, 1) ⊕ C∞0 (0, 1),

since the adjoint operator is invariant under closure. One concludes using stan-dard calculations ([33]) that the adjoint operator is determined by the same oper-ator valued matrix (2) on the set of functions satisfying the following five condi-tions


(1) U = (u1, u2) ∈ L2[0, 1] ⊕ L2[0, 1]; (10)

(2) u1 ∈ W 12 (ε, 1) for any 0 < ε < 1; (11)

(3) The function

ωU(x) := −ρ(x)u′1(x)+β(x)

xu2(x) (12)

is absolutely continuous on [0, 1];(4)

d

dxωU(x) = d

dx

(−ρ(x) d

dxu1 + β(x)

xu2

)∈ L2[0, 1]; (13)

(5) −β(x)x

d

dxu1 + m

x2u2 ∈ L2[0, 1]. (14)

The function ωU is called transformed derivative� and is well-defined for anyfunction

U = (u1, u2)), u1 ∈ W 12,loc(0, 1) ∩ L2[0, 1], u2 ∈ L2[0, 1].

The transformed derivative appearing in the boundary conditions for the matrixdifferential operator L plays the same role as the usual derivative for the stan-dard one-dimensional Schrödinger operator. The function ωU corresponding toU ∈ Dom(L∗) belongs to W 1

2 (0, 1), since it is absolutely continuous and (13)holds.

Let us calculate the sesquilinear boundary form of the adjoint operator. Thisform can be used to describe all self-adjoint extensions of Lmin as restrictionsof the adjoint operator to Lagrangian planes with respect to this form. Let U ,V ∈ Dom(L∗

min), then integrating by parts we get

〈L∗minU,V 〉 − 〈U,L∗

minV 〉

=⟨

d

dx

(−ρu′1 +

β

xu2

), v1

⟩+⟨−βx

d

dxu1 + m

x2u2, v2

⟩−

−⟨u1,

d

dx

(−ρv′1 +

β

xv2

)⟩−⟨u2,−β

x

d

dxv1 + m

x2v2

⟩

= limε↘0,τ↗1

{∫ τ

ε

(d

dxωU

)v1 dx +

∫ τ

ε

(−βxu′1 +

m

x2u2

)v2 dx −

−∫ τ

ε

u1

(d

dxωV

)dx −

∫ τ

ε

u2

(−βxv1

′ + m

x2v2

)dx

}

= limε↘0,τ↗1

{ωU(x)v

′1(x)|τx=ε −

∫ τ

ε

ωUv1 dx −∫ τ

ε

β

xu′1v2 dx −

� The transformed derivative is a generalization of the quasi-derivatives described, for example,by W. N. Everitt, C. Bennewitz and L. Markus [6, 7].


− u1(x)ωV (x)|τx=ε +∫ τ

ε

u′1ωV dx +∫ τ

ε

u2β

xv1

′ dx}

= limε↘0,τ↗1

{ωU(x)v1(x)|τx=ε − u1(x)ωV (x)|τx=ε}. (15)

Note that the limits in the last formula cannot be always substituted by the limitvalues of the functions, since the functions u1 and v1 are not necessarily bounded atthe origin. On the other hand the limit as τ ↗ 1 can be calculated using continuityof all four functions at the regular endpoint x = 1. This boundary form will beused to determine the deficiency indices of the operator Lmin and describe its self-adjoint extensions. This method of using boundary forms to describe self-adjointextensions of symmetric operators is classical and is well described for example in[3] (vol. 2) and [33].

THEOREM 4.1. The operator Lmin is a symmetric operator in the Hilbert spaceH with finite equal deficiency indices.

(1) If the operator matrix L is singular quasiregular (i.e. quasiregularity condi-tions are satisfied and m(0) = 0), then the deficiency indices of Lmin are equalto (1, 1) and all self-adjoint extensions of Lmin are described by the standardboundary condition

ωU(1) = h1u1(1), h1 ∈ R ∪ {∞}. (16)

(2) If the operator matrix is regular or is not quasiregular then the deficiencyindices of Lmin are equal to (2, 2). The self-adjoint extensions of Lmin are de-scribed by pair of boundary conditions using the following alternatives coveringall possibilities:

(a) If ρ(0)m(0) − β2(0) = 0 or β(0) = 0, then the first component u1 of anyvector from the domain of the adjoint operator L∗

min is continuous on theclosed interval [0, 1]. All local � self-adjoint extensions of the operator Lmin

are described by the standard boundary conditions ��

ωU(1) = h1u1(1), ωU(0) = h0u1(0), h0,1 ∈ R ∪ {∞}. (17)

(b) If

ρ(0)m(0)− β2(0) = 0,d

dx(ρm− β2)(0) = 0, and β(0) = 0,

then the first component u1 of any vector from the domain of the adjointoperator L∗

min admits the asymptotic representation

u1(x) = kwU(0) ln x + cU + o(1), as x → 0, (18)� The family of all self-adjoint extensions of Lmin can easily be described using our analysis. The

corresponding formulas are not written here only in order to make the presentation more transparent.�� In the case hα = ∞, α = 0, 1 the corresponding boundary condition should be written asu1(α) = 0 or cU = 0.


where

k = −β2(0)

ρ(0)

1d

dx (ρm− β2)|x=0

and cU is an arbitrary constant depending on U . Then all local self-adjointextensions of the operator Lmin are described by the nonstandard boundaryconditions ��

ωU(1) = h1u1(1), ωU(0) = h0cU , h0,1 ∈ R ∪ {∞}. (19)

Information concerning the deficiency indices of Lmin and self-adjoint localboundary conditions is collected in Table I.

Proof. In order to describe all local boundary conditions the points x = 0 andx = 1 can be considered separately. The point x = 1 is a regular boundary point,since the functions ρ−1, β/x,m/x2 are infinitely differentiable in a neighborhoodof this point. The symmetric boundary condition at the point x = 1 can be writtenin the form

ωU(1) = h1u1(1), (20)

where h1 ∈ R ∪∞ is a real constant parametrizing all symmetric conditions (see[43] and Case C below for details). The extension of the operator Lmin to the setof infinitely differentiable functions with support separated from the origin andsatisfying condition (20) at the point x = 1 will be denoted by Lh1 .

Let us study the deficiency indices of the operator Lh1 . The operator adjoint toLh1 is the restriction of L∗

min to the set of functions satisfying (20). This operatoris defined by the operator matrix with real coefficients, therefore the deficiency

Table I.

ρ(0)m(0)− β2(0) = 0 ρ(0)m(0)− β2(0) = 0

ddx (ρm− β2)|x=0 = 0 d

dx (ρm− β2)|x=0 = 0

A B C

β(0) = 0 indices (2,2) indices (2,2) indices (2,2)

2 standard b.c. (17) 2 standard b.c. (17) 2 standard b.c. (17)

β(0) = 0 indices (2,2) indices (2,2) indices (1,1)

2 standard b.c. (17) 2 nonstandard b.c. (19) 1 standard b.c. (16)

The letters A, B, and C refer to the three cases considered in the proof of the theorem.


indices of Lh1 are equal. Moreover, the differential equation on the deficiencyelement gλ for any λ /∈ R [3] is given by

d

dx

(−ρ(x) d

dxgλ1 +

β(x)

xgλ2

)+ q(x)gλ1 = λgλ1 ,

−β(x)x

d

dxgλ1 +

m(x)

x2gλ2 = λgλ2 ;

(21)

and it can be reduced to the following scalar differential equation for the firstcomponent

− d

dx

(ρ(x)+ β(x)

x

1

λ−m(x)/x2

β(x)

x

)d

dxgλ1 + q(x)gλ1 = λgλ1 . (22)

The component gλ2 can be calculated from gλ1 using the formula

gλ2 = − 1

λ−m(x)/x2

β(x)

x

d

dxgλ1 .

Equation (22) is a second-order ordinary differential equation with continuouslydifferentiable coefficients. Since the principle coefficient in this equation for non-real λ is separated from zero on the interval (ε, 1], the solutions are two timescontinuously differentiable functions (18).

Boundary condition (20) implies that the first component satisfies the boundarycondition at point x = 1

−(ρ(1)+ β2(1)

λ−m(1))

d

dxgλ1 (1) = h1g

λ1 (1). (23)

This condition is nondegenerate, since λ is nonreal. Therefore the subspace ofsolutions to Equation (21) satisfying condition (20) has dimension 1. But thesesolutions do not necessarily belong to the Hilbert space H = L2[0, 1] ⊕ L2[0, 1].If the nontrivial solution is from the Hilbert space, gλ ∈ H , then the operator Lh1 issymmetric with deficiency indices (1, 1). Otherwise the operator Lh1 is essentiallyself-adjoint ([42]). If the principal coefficient of Equation (22) is bounded andseparated from zero on the interval [0, 1], then gλ ∈ H and the operator Lh1 hasdeficiency indices (1,1). The last condition is satisfied if for example m(0) = 0and ρ(0)m(0) − β2(0) = 0, since �λ = 0. Complete analysis of Equation (22)can be carried out using WKB method ([34]). We are going instead to analyze theboundary form.

Let us study the singular point x = 0 in more detail. We are going to considerthe following three possible cases:

(A) The first quasiregularity condition (8) is not satisfied.(B) The first quasiregularity condition is satisfied, but the second quasiregularity

condition (8) is not satisfied.(C) The quasiregularity conditions (8) are satisfied.


The case C includes the set of regular operator matrices.

Case A. Consider arbitrary cutting function ϕ ∈ C∞[0, 1] equal to 1 in a certainneighborhood of the origin and vanishing in a neighborhood of the point x = 1.The function

W = (m(0)xϕ(x), β(0)xϕ(x))

obviously belongs to the domain of the adjoint operator L∗h1, since the support of

the functionW is separated from the point x = 1 and condition (20) is therefore sat-isfied. The functionW is not identically equal to zero, since the first quasiregularitycondition (8) is not satisfied.

Consider arbitrary U ∈ Dom (L∗min). Then formula (15) implies that the limit

limε↘0

{−ωU(ε)w1(ε)+ u1(ε)ωW(ε)}exists. Taking into account that

ωU is absolutely continuous on the interval [0, 1];limε↘0w1(ε) = 0;limε↘0 ωW(ε) = −ρ(0)m(0)+ β2(0) = 0;

we conclude that the limit u1(0) = limε↘0 u1(ε) exists for arbitrary function U ∈Dom (L∗). Hence, the boundary form of the operator L∗

h1is given by

〈L∗h1U,W 〉 − 〈U,L∗

h1W 〉 = −ωU(0)w1(0)+ u1(0)ωW(0),

and is not degenerate. The operator L(h1) has deficiency indices (1,1), and allsymmetric boundary conditions at the point x = 0 are standard

ωU(0) = h0u1(0). (24)

Case B. Let us introduce the following notation

c0 = d

dx(ρ(x)m(x)− β2(x))|x=0 = 0. (25)

In addition we suppose that β(0) = 0. To prove that the boundary form is notdegenerate (and hence the deficiency indices of Lh1 are (1, 1)) consider the twovector functions

F = 1 +

∫ x

0

β(t)

ρ(t)dt

x

, (26)

G = −

∫ 1

x

(c0

ρ(0)β(0)+ β(t)

tρ(t)

)dt

1

. (27)


Multiplying the functions F and G by the scalar function ϕ introduced above onegets functions from the domain of the operator L∗

h1. The fact that these functions

satisfy (10), (11), (13), (14) is a result of straightforward calculations. We have

ωF (ε) ≡ 0, limε↘0

f1(ε) = 1,

and

ωG(ε) = − c0

β(0)ρ(0)ρ(ε), g1(ε) = β(0)

ρ(0)(ln ε)+ cG + o(1).

Hence the boundary form of L∗h1(h1) calculated on ϕF and ϕG is given by

〈L∗h1ϕG, ϕF 〉 − 〈ϕG,L∗

h1ϕF 〉 = c0

β(0)= 0.

Therefore the deficiency indices of Lh1 are equal to (1,1).Let us prove that the asymptotic representation (18) holds for any function V

from the domain of the operator adjoint to Lmin. Consider the boundary form of theadjoint operator calculated on the function V and the above introduced function G.The following limits obviously exist

∃ limε↘0

[−ωG(ε)v1(ε)+ g1(ε)ωV (ε)]

= limε↘0

[−(− c0

β(0)+ o(

√ε)

)v1(ε)+

+(β(0)

ρ(0)ln ε + cU + o(1)

)(ωV (0)+ o(

√ε))

]

⇒ ∃ limε↘0

[c0

β(0)(1 + o(

√ε))v1(ε)+ β(0)

ρ(0)ωV (0) ln ε

].

It follows that (18) holds. The parameters ωU(0) and cU are independent, when Uruns over Dom(L∗

h1). This follows easily from the fact that the function (u1, u2) =

(1, 0) belongs to the domain of L∗min.

Substituting the asymptotic representation (18) for arbitrary U,V ∈ Dom(Lh1)

into the boundary form

〈L∗h1U,V 〉 − 〈U,L∗

h1V 〉 = lim

ε↘0(−ωU(ε)v1(ε)+ u1(ε)ωV (ε))

=−ωU(0)cV + cUωV (0).Hence all local self-adjoint extensions are described by nonstandard boundaryconditions (19).

To complete the study of Case B, let β(0) = 0. Consider the function F given by

(26) and the function S= ( x0

). Then the boundary form calculated on the vectors

ϕF and ϕS is nondegenerate

〈L∗h1ϕS, ϕF 〉 − 〈ϕS,L∗

h1ϕF 〉 = ρ(0) = 0,


and therefore the operator Lh1 has deficiency indices (1, 1). Let us prove that thecomponent u1 of any vector from the domain of the adjoint operator is continuousin the closed interval. Note that

ωS(x) = −ρ(x) and s1(0) = 0.

Consider the boundary form of L∗h1

calculated on ϕS and arbitrary V ∈ Dom (Lh1)

〈Lh1ϕS, V 〉 − 〈ϕS,Lh1V 〉 = limε↘0(ρ(ε)v1(ε)+ εωV (ε))

= − limε↘0

ρ(ε)v1(ε).

Since ρ(0) is not equal to zero, the limit limε↘0 v1(ε) exists and therefore self-adjoint boundary conditions can be written in the standard form (17) as in Case A.This completes investigation of Case B.

Case C. Suppose in addition that β(0) = 0. It follows that the matrix is singularquasiregular. Consider the vector function

E = −

∫ 1

x

β(t)

tρ(t)dt

1

,

which belongs to the domain of the adjoint operator L∗min due to quasiregular

conditions. Therefore ϕE ∈ Dom(L∗h1). Then for any function U ∈ Dom(L∗

h1) the

boundary form is given by

〈L∗h1U, ϕE〉 − 〈U,L∗

h1ϕE〉 = − lim

ε↘0ωU(ε)e1(ε),

since ωE(ε) ≡ 0. Note that e1 diverges to infinity due to our assumption β(0) = 0

v1(ε) ∼ε↘0β(0)

ρ(0)ln ε → ∞.

Since the limit limε↘0 ωU(ε) exists it should be equal to zero

ωU(0) = 0. (28)

Hence taking into account that ωU ∈ W 12 [0, 1] one concludes that

ωU(ε) = o(√ε). (29)

On the other hand, condition (13) implies that

xd

dxu1 = β

ρu2 − x

ρωU ∈ L2[0, 1]. (30)

It follows from Cauchy inequality that

u1(ε) = O

(1√ε

). (31)


Formulas (29) and (31) imply that the boundary form is identically equal to zero.Therefore the operator L(h1) is essentially self-adjoint in this case. (Note that eachfunction from the domain of arbitrary self-adjoint extension of Lmin automaticallysatisfies the boundary condition (28) at the singular point.)

To accomplish the investigation of Case C, assume β(0) = 0. The first quasireg-ularity condition (8) implies that m(0) = 0. The second quasiregularity condition(8) implies then that (d/dx)m|x=0 = 0. It follows that point zero is a regular pointfor the operator matrix L. Therefore the deficiency indices of L(h1) are equal to(1, 1) and the local self-adjoint extensions are described by standard boundary con-ditions ([17, 43]). We have already proven this result. Indeed taking into accountthat u1 ∈ W 1

2 (0, 1) and that the function ω(ε) is absolutely continuous the abovementioned fact follows immediately from (15). This accomplishes the investigationof Case C. The theorem is proven. ✷COROLLARY 4.1. The theorem implies that the operator Lh1 is essentially self-adjoint if and only if the operator matrix is singular quasiregular. Otherwise it hasdeficiency indices (1,1).

Nonstandard boundary conditions (19) at the singular point described by The-orem 4.1 are similar to the boundary conditions appearing in the studies of one-dimensional Schrödinger operator with Coulomb potential

− d2

dx2− γ

xin L2(R).

In what follows we are going to study the essential spectrum of the self-adjointextensions of the operator Lmin. Since the deficiency indices of this operator arealways finite, the essential spectrum does not depend on the particular choice ofthe boundary conditions. The same holds true for nonlocal boundary conditionsand therefore our restriction to the case of local boundary conditions can be waived.Therefore in the course of the paper we are going to denote by L some self-adjointextension of the minimal operator.

5. Transformation of the Operator

In the current section we are going to transform the self-adjoint operator L toanother self-adjoint operator acting in the Hilbert space H = L2[0,∞)⊕L2[0,∞).The reason to carry out this transformation is pure technical – we would like to beable to use Fourier transform.

Consider the following change of variables

x = e−y , dx = −e−y dy = −x dy, (32)

mapping the interval [0,∞) onto the interval [0, 1] and the corresponding unitarytransformation between the spaces L2[0, 1] and L2[0,∞)

.: ψ(x) %→ ψ(y) = ψ(e−y)e−y/2. (33)


The points 0 and ∞ are mapped to 1 and 0, respectively, and the following formulaholds∫ 1

0‖ ψ(x) ‖2 dx =

∫ ∞

0‖ ψ(e−x) ‖2 e−y dy.

The inverse transform is given by

.−1: ψ(y) %→ ψ(x) = 1√xψ(−ln x). (34)

To determine the transformed operator denoted by K let us calculate the trans-formed operator matrix first componentwise

K11:

√x

([− d

dxρ

d

dx+ q(x)

]1√xψ(−ln x)

)

= √x

(− d

dxρ

[1

2x3/2ψ(−ln x)+ 1

x3/2ψ ′(−ln x)

])+ q(x)ψ(−ln x)

= √x

(ρ ′x

(1

2x3/2ψ(−ln x)+ 1

x3/2ψ ′(−ln x)

)+

+ ρ[− 3

4x5/2ψ(−ln x)+ 1

2x3/2ψ ′(−ln x)

−1

x+ −3

2x5/2ψ ′(−ln x)+

+ 1

x3/2ψ ′′(−ln x)

−1

x

])+

+ q(x)ψ(−ln x)

= − ρ

x2ψ ′′(−ln x)+

(ρ ′xx

− 2ρ

x2

)ψ ′(−ln x)+

(ρ ′x2x

− 3

4

ρ

x2

)ψ(−ln x)+

+ q(x)ψ(−ln x)

= − d

dy

ρ

x2

d

dyψ(−ln x)+

(q(x) + ρ ′x

2x− 3ρ

4x2

)ψ(−ln x).

K12:

√x

(d

dx

β

x

1√xψ(−ln x)

)

= √x

d

dx

(β

x3/2ψ(−ln x)

)

= − βx2ψ ′(−ln x)+ ψ(−ln x)

(β ′xx

− 3β

2x2

)


= − d

dy

(β

x2ψ(−ln x)

)− x

(β ′xx2

− 2β

x3

)ψ(−ln x)+

(β ′xx

− 3β

2x2

)ψ(−ln x)

= − d

dy

(β

x2ψ

)+ β

2x2.

K21 is the conjugated expression to K12

β

x2

d

dy+ 1

2

β

x2.

K22: m/x2.

Finally the transformed operator matrix will be denoted by K and it is given by

K =

− d

dy

ρ

x2

d

dy+(q(x)+ ρ ′x

2x− 3ρ

4x2

)− d

dy

β

x2+ β

2x2

β

x2

d

dy+ 1

2

β

x2

m

x2

:=

(A C∗C D

). (35)

To define a self-adjoint operator corresponding to this operator matrix one hasto consider first the minimal operator Kmin being the closure of the differentialoperator given by (35) on the domain of functions from C∞

0 [0,∞) ⊕ C∞0 [0,∞).

Then one has to study the deficiency indices of this operator and describe all itsself-adjoint extensions. This analysis is equivalent to the one carried out in theprevious section for the operator Lmin. The self-adjoint extensions of the operatorsLmin and Kmin are in one-to-one correspondence given by the unitary equivalence(33), (34). Therefore we conclude that the deficiency indices of the operator Kmin

are equal and finite ((1, 1) or (2, 2) depending on the properties of the coefficients).Let us denote by K one of the self-adjoint extensions of the minimal operator. Theessential spectrum of the operator will be studied. The analysis does not dependon the choice of self-adjoint extension, since the deficiency indices of the minimaloperator are finite.

It is easier to study pseudodifferential operators on the whole axis instead ofthe half axis. The reason is that the manifold [0,∞) has nontrivial boundary andtherefore even the momentum operator cannot be defined as a self-adjoint operatorin L2[0,∞). It appears more convenient for us to study the corresponding problemon the whole real line in order to avoid these nonessential difficulties related to theboundary point y = 0. In this way the problem of studies of the matrix differentialoperator can be reduced to a certain pure algebraic problem.

Consider the Hilbert space H = L2(R) ⊕ L2(R). The operator K acting in H

can be chosen in such a way that its essential spectrum coincides with the essentialspectrum of the operator K.

In order to simplify the discussion of the essential spectrum we have to chosespecial continuation of the operator. However, this program applied to the operatorKmin itself meets some difficulties and it appears more convenient for us to perform


this program on a later stage of the investigation of the operator, namely during thestudies of the cleaned resolvent of the operator.

6. Resolvent Matrix and the Hain–Lüst Operator

The resolvent of the operator K will be used to study its essential spectrum. Thedifference between the resolvents of any two self-adjoint extensions of the minimaloperator Kmin is a finite rank operator and it follows that the essential spectrum isindependent of the chosen self-adjoint extension. In fact it is enough to calculatethe resolvent of the operator K on any subspace of finite codimension, for exampleon the range of the minimal operator Kmin. We are going to consider the resolventequation

(Kmin − µ)−1F = U,

for µ satisfying one of the following two conditions

(i) (µ = 0;(ii) µ ∈ R, |µ| ) 1.

Formula (36) below shows that resolvent’s denominator T (µ) has no additionalsingularities outside x = 0 for all nonreal values of the parameter µ. For suffi-ciently large real µ the same holds true if either m(0) = 0, or m(0) = 0, thequasiregularity conditions (8) hold and

sign µ sign m(0+) = −1.

If the quasiregularity conditions hold then m(0+) � 0 and the parameter µ canalways be chosen to be small negative, µ* −1.

For F ∈ R(Kmin) and U ∈ C∞0 [0,∞)⊕ C∞

0 [0,∞) the resolvent equation canbe written as follows

f1 = (A− µ)u1 + C∗u2, f2 = Cu1 + (D − µ)u2.

Using the fact that the operator (D−µ) is invertible for nonreal µ one can calculateu2 from the second equation

u2 = (D − µ)−1f2 − (D − µ)−1Cu1

and substitute it into the first equation to get

f1 = ((A− µ)− C∗(D − µ)−1C)u1 + C∗(D − µ)−1f2.

The last equation can easily be resolved using Hain–Lüst operator, which is anal-ogous to the regularized determinant of the matrix K

T (µ) = (A− µI)− C∗(D − µI)−1C

= − d

dy

(ρ

x2− β2

x2(m− µx2)

)d

dy− µ+

+{q(x) + ρ ′x

2x− 3ρ

4x2− β2

4x2(m− µx2)− x d

dx

(β2

2x2(m− µx2)

)}. (36)


Elementary calculations show that under quasiregular conditions (8) both coef-ficients in the expression above are smooth and bounded. The principle coefficient

ρ

x2− β2

x2(m− µx2)

is uniformly separated from zero. We consider this operator for µ * −1 on theset C∞

0 [0,∞) and use the same notation for its Friedrichs extension described bythe Dirichlet boundary condition at the origin. This operator has been introducedin a special case by K. Hain and R. Lüst during the investigation of problemsof magnetohydrodynamics. In what follows we are going to show that Hain–Lüstoperator plays the key role in the investigation of the essential spectrum.

The role of the quasiregularity conditions for the Hain–Lüst operator is ex-plained by the following lemma.

LEMMA 6.1. Let µ /∈ Rangex∈[0,1]((m(x))/x2), then the coefficients of the Hain–Lüst operator (36)

f (x) = ρ

x2− β2

x2(m− µx2),

and

g(x) = q(x) + ρ ′x2x

− 3ρ

4x2− β2

4x2(m− µx2)− x d

dx

(β2

2x2(m− µx2)

)− µ,

are uniformly bounded functions if and only if the quasiregularity conditions (8)hold.

Comment. The condition µ /∈ Rangex∈[0,1]((m(x))/x2) holds, for example, if theparameter µ either nonreal or µ ∈ R, µ* −1.

Proof. Let the quasiregularity conditions (8) be satisfied. Then the coefficient

f (x) = ρm− β2 − µx2

x2(m− µx2)

is uniformly bounded, since by (8)

ρ(x)m(x) − β2(x) ∼x→0 cx2

and the factor m− µx2 is uniformly separated from 0. The function

g(x)− q(x) + µ+ f (x)

4

= ρ ′x2x

− ρ

x2− x

(β2

2x2(m− µx2)

)′

x


= ρ ′x2x

− ρ

x2− x

(β2 − ρµ

2x2(m− µx2)

)′

x

− x(

ρµ

2x2(m− µx2)

)′

x

= −x(

β2 − ρµ2x2(m− µx2)

)′

x

+ µx(

ρx2

2x2(m− µx2)

)′

x

is also uniformly bounded.On the other hand, the boundedness of the leading coefficient

f (x) = ρm− β2 − µx2

x2(m− µx2)

implies conditions (8) under the assumptions of the lemma. The lemma is proven. ✷Similar result has been proven for magnetohydrodynamic operator in [17].The resolvent matrix can be presented by

M(µ) ≡ (Kmin − µ)−1

=(

T−1(µ) −T−1(µ)[C∗(D − µI)−1]−[(D − µI)−1C]T−1(µ) (D − µI)−1 + [(D − µI)−1C]T−1(µ)[C∗(D − µI)−1]

). (37)

The last expression determines the resolvent of any self-adjoint extension K ofthe minimal operator Kmin on the subspace R(Kmin) which has finite codimen-sion. Therefore this resolvent matrix determines the essential spectrum of anyself-adjoint extension K. In order to calculate the essential spectrum we are goingto consider perturbations of the calculated resolvent by compact operators. This isdiscussed in the following section.

7. The Asymptotic Hain–Lüst Operator

The essential spectra of two operators coincide if the difference between theirresolvents is a compact operator. This idea of relatively compactness was usedin applications to magnetohydrodynamics by T. Kako [22]. Even if the expressionfor the resolvent is much more complicated than the one for operator itself weprefer to handle with the resolvent. We are going to simplify the expression forthe resolvent step by step using Weyl theorem. We call this procedure cleaning ofthe resolvent. Therefore we are going to perturb the resolvent operator M(µ) bycompact operators in order to simplify it. Our aim is to factorize the pseudodiffer-ential operator M(µ) into a sum of two pseudodifferential operators with symbolsdepend on the coordinate and momentum, respectively. In our calculations we aregoing to use the Calkin calculus [13]. We say that any two operators A and B areequal in Calkin algebra if their difference is a compact operator. The followingnotation for the equivalence relation in Calkin algebra will be used throughout the


paper: A =B. Since all operators appearing in the decomposition (37) are in factpseudodifferential the following notation for the momentum operator will be used

p = 1

i

d

dy. (38)

This symbol will denote the differential expression in the first half of this section.The same notation will be used for the symbol of the pseudodifferential operatoron the real line in the rest of the paper.

Let us introduce the asymptotic Hain–Lüst operator for the generic casem(0) = 0

Tas(µ) = a(µ)

(− d2

dy2+ c(µ)

)≡ a(µ)(p2 + c(µ)), (39)

where

a(µ) = limx→0

(ρ

x2− β2

x2(m− µx2)

)= l0 − µρ(0)

m(0),

l0 = limx→0

(ρ − β2

m

x2

), (40)

c(µ)= 1

4− µ

a(µ).

The domain of the asymptotic Hain–Lüst coincides with the set of functions fromthe Sobolev space W 2

2 satisfying the Dirichlet boundary condition at the origin:{ψ ∈ W 2

2 ([0,∞)), ψ(0) = 0}. We obtain the asymptotic Hain–Lüst operator bysubstitution the coefficients of the second-order differential Hain–Lüst operator bytheir limit values at the singular point. It will be shown that the additional branch ofessential spectrum of L is determined exactly by the symbol of asymptotic Hain–Lüst operator.

To prove that the difference between the inverse Hain–Lüst and inverse asymp-totic Hain–Lüst operators is compact we are going to use Lemma B.4. We decidedto devote a separate appendix to this lemma which is of special interest in the theoryof pseudodifferential operators (see Appendix B, where the proof of this lemmacan be found). This lemma implies that the difference of the inverse Hain–Lüstoperators is compact

T −1(µ)− T −1as (µ) ∈ S∞ (41)

for sufficiently large |µ| to guarantee the invertibility of the both operators. Notethat both operator functions −T −1(µ) and −T −1

as (µ) are operator valued Herglotzfunctions ([32]).

8. Cleaning of the Resolvent

This section is devoted to the cleaning of the resolvent, which is based on formula(41). The main algebraic tool is Calkin calculus ([13]) and Appendix B.


Using Calkin algebra and Lemma B.1 formula (41) can be almost rigorouslywritten as follows

p T −1(µ)p = 1ρ

x2 − β2

x2(m−µx2)

. (42)

In fact to apply Lemma B.1 one needs extra regularizator h – any bounded van-ishing at infinity function (see formula (80)). The operator p T −1(µ)p here is theclosure of the bounded operator defined originally on W 1

2 [0,∞). Let us introducethe function

b(x, µ) = β

m− µx2. (43)

Our aim is to find a matrix differential operator equivalent in Calkin algebra tothe operator M(µ) given by (37). Using (41) and the fact (the result of straight-forward calculations) that the operators C∗(D − µI)−1 and (D − µI)−1C underquasiregular conditions are first order differential operators with bounded smoothcoefficients we obtain �

M(µ) =

1a(µ)

1p2+c(µ) − b(0,µ)

a(µ)

ip+1/2p2+c(µ)

− b(0,µ)a(µ)

−ip+1/2p2+c(µ)

x2

m−µx2 + [(D − µI)−1C]T−1(µ)[C∗(D − µI)−1]

. (44)

The expressions (±ip + 1/2)/(p2 + c(µ)) are considered as bounded opera-tors defined on L2[0,∞) by

(±ip + 1/2)(p2 + c(µ))−1,

where (p2 + c(µ))−1 is the resolvent of the Laplace operator p2 with the Dirichletboundary condition at the origin. Substituting expressions for the operators C andD from (35) we get

M(µ) =( 1

a(µ)1

p2+c(µ) − b(0,µ)a(µ)

ip+1/2p2+c(µ)

− b(0,µ)a(µ)

−ip+1/2p2+c(µ)

x2

m−µx2 + b(x,µ)(−ip+ 1/2)T −1(µ)(ip + 1/2)b(x, µ)

).

Let us concentrate our attention to the element (22). We consider this differentialoperator on the set W 1

2 [0,∞).b(x, µ)(−ip + 1/2)T −1(µ)(ip + 1/2)b(x, µ)

= b(x, µ)(−ip + 1/2)T −1as (µ)(ip + 1/2)b(x, µ) +

+ b(x, µ)(−ip + 1/2)T −1(µ)(Tas(µ)− T (µ))T −1as (µ)(ip + 1/2)b(x, µ)

= b(x, µ)p2 + 1/4

a(µ)(p2 + c(µ))b(x, µ)+� In fact only the condition m(0) = 0 is used here. This relation follows from the first

quasiregularity condition (8).


+ b(x, µ)(−ip + 1/2)T −1(µ)

[d

dy

(ρ

x2− β2

x2(m− µx2)− a(µ)

)d

dy

]×

× ip + 1/2

a(µ)(p2 + c(µ))b(x, µ).

The last equality in Calkin algebra holds due to the following observations:

(1) The operator T −1as (µ)(ip + 1/2) is bounded.

(2) Since the minor terms in both T (µ) and Tas(µ) are bounded functions,Lemma B.3 and (1) imply that the following operator is compact

(−ip + 1/2)T −1(µ) {bounded function tending to 0 at infinity}= (−ip + 1/2)T −1

as (µ) {bounded function tending to 0 at infinity}= 0.

To transform the first term the following equality has been used

(−ip + 1/2)T −1as (µ)(ip + 1/2) = p2 + 1/4

p2 + c(µ) .

Using

b(x, µ)p2 + 1/4

p2 + c(µ)b(x, µ) = b(0, µ)p2 + 1/4

p2 + c(µ)b(0, µ)

(b ∈ L∞[0,∞) and has limit at ∞, Lemma 6.1 from [17]), we get

b(x, µ)(−ip + 1/2)T −1(µ)(ip + 1/2)b(x, µ)

= b2(x, µ)

a(µ)+ b2(0, µ)

a(µ)

1/4 − c(µ)p2 + c(µ) +

+ b(x, µ)(−ip + 1/2)T −1(µ)

[−p

(ρ

x2− β2

x2(m− µx2)− a(µ)

)]×

× ip2 + p/2a(µ)(p2 + c(µ))b(x, µ).

The operator

ip2 + p/2a(µ)(p2 + c(µ))b(x, µ) ≡ (ip2 + p/2)T −1

as (µ)b/x,µ)

is bounded. Consider the operator

b(x, µ)(−ip + 1/2)T −1(µ)

[−p

(ρ

x2− β2

x2(m− µx2)− a(µ)

)]

= b(x, µ) 1ρ

x2 − β2

x2(m−µx2)

(ρ

x2− β2

x2(m− µx2)− a(µ)

)


due to Lemma B.1 and the equality following from (8)(ρ

x2− β2

x2(m− µx2)− a(µ)

)|x=0 = 0. (45)

Lemma B.1 could be applied here, since one can easily that the operator

b(x, µ)(1/2)T −1(µ)

[−p

(ρ

x2− β2

x2(m− µx2)− a(µ)

)]

is compact.Therefore the element (22) is equivalent in Calkin algebra to the following

operator

x2

m− µx2+ b2(x, µ)

a(µ)+ b2(0, µ)

a(µ)

1/4 − c(µ)p2 + c(µ) −

− b(x, µ) 1ρ

x2 − β2

x2(m−µx2)

(ρ

x2− β2

x2(m− µx2)− a(µ)

)1

a(µ)b(x, µ).

The following formula for the cleaned resolvent matrix has been obtained

M(µ)

=

1

a(µ)

1

p2 + c(µ) − b(0, µ)a(µ)

ip + 1/2

p2 + c(µ)

− b(0, µ)a(µ)

−ip + 1/2

p2 + c(µ)x2

m− µx2+ b2(0, µ)

a(µ)

1/4 − c(µ)p2 + c(µ) + b2(x, µ)

ρ

x2 − β2

x2(m−µx2)

. (46)

Let us remind that the formal expression

1

a(µ)

1

p2 + c(µ)in all four matrix entries denotes the resolvent of the asymptotic Hain–Lüst opera-tor.

The last matrix can be written (at least formally) as a sum of two matricesdepending on x and p only: M(µ) =X(x)+ P(p), where

X(x) =

0 0

0x2

m− µx2+ b2(x, µ)

ρ

x2 − β2

x2(m−µx2)

,

P (p) =

1

a(µ)

1

p2 + c(µ) −b(0, µ)a(µ)

ip + 1/2

p2 + c(µ)−b(0, µ)a(µ)

−ip + 1/2

p2 + c(µ)b2(0, µ)

a(µ)

1/4 − c(µ)p2 + c(µ)

.


In Section 4, to handle pseudodifferential operators, we discussed the extensionof all operators to certain operators acting in the Hilbert space H = L2(R) ⊕L2(R) ⊃ L2[0,∞) ⊕ L2[0,∞). This procedure can easily be carried out for thecleaned resolvent. Let us continue all involved functions b(x(y), µ), ρ(x(y)) andm(x(y)) to the whole real line as even functions of y. Consider the operator gen-erated by the continued matrix symbol X(x(y))+ P(p). This operator is boundedoperator defined on the whole Hilbert space H. The essential spectrum of the newoperator coincides (without counting multiplicity) with the essential spectrum ofthe original operator M(µ). Really Glazman’s splitting procedure ([3]) and Weyltheorem on compact perturbations ([24]) imply that the essential spectrum of thenew operator coincides with the union of the essential spectra of the two operatorsgenerated by the operator matrix on the two half-axes:

1

p2 + c(µ) |L2(R) =1

p2 + c(µ) |L2(−∞,0] ⊕1

p2 + c(µ) |L2[0,∞),

where1

p2 + c(µ) |L2(−∞,0] and1

p2 + c(µ) |L2[0,∞)

denote the resolvents of the Laplace operator p2 on the corresponding semiaxiswith the Dirichlet boundary condition at the origin. In the last formula p denotesthe momentum operator in the left-hand side and the differential expression in theright one.

One can easily prove that the unitary transformation(f1(y)

f2(y)

)%→(f1(−y)−f2(−y)

)relates the matrix operators generated in the orthogonal decomposition of theHilbert space

H = (L2(−∞, 0] ⊕ L2(−∞, 0])⊕ (L2[0,∞)⊕ L2[0,∞)).Hence, the two operators appearing in this orthogonal decomposition are unitaryequivalent and therefore have the same essential spectrum.

The problem of calculation of the essential spectrum has been transformed to apure algebraic problem.

9. Calculation of the Essential Spectrum

In order to apply Proposition A.1 from Appendix A let us introduce two matrixoperator functions

Q =

0 0

0ρ(0)

m(0)a(µ)

(47)


and

Y (y) =

0 0

0x2

m− µx2+ b2(x, µ)

ρ

x2 − β2

x2(m−µx2)

− ρ(0)

m(0)a(µ)

. (48)

Let us remind the reader that everywhere in the paper x is considered as a functionof the variable y, x = e|y|, where we have taken into account the even continuationof all parameters of the matrix for negative values of y. The matrices Q, Y (y) andP(p) satisfy the conditions of Proposition A.1. In addition, the matrix functionsY (y) and P(p) are continuous on the real line and have zero limits at infinity.All matrix functions are depending on the parameter µ. Therefore the essentialspectrum of the resolvent operator M(µ) is given by (72)

σess(M(µ)) = σess(Q + P) ∪ σess(Q + Y).

To calculate the essential spectra of the operators Q + P and Q + Y we use thefact that the determinants of the corresponding matrices Q+ P(p) and Q+ Y (y)are equal to zero identically. It follows that one of the two eigenvalues of the eachmatrix is identically zero. Therefore the essential spectra of the operators coincideswith the range of the second (nontrivial) eigenvalues when y resp.p runs over thewhole real axis. This simple fact is a result of straightforward calculations. Thenontrivial eigenvalues coincide with the traces of the corresponding 2× 2 matricesQ+ P(p) and Q+ Y (y). The trace of the matrix M(µ) is given by

Tr (M(µ)) = Tr (Y (y))+ Tr (P (p))− Tr (Q)

= 1

a(µ)

1

p2 + c(µ) +x2

m− µx2+

+ b2(0, µ)

a(µ)

1/4 − c(µ)p2 + c(µ) + b2(x, µ)

ρ

x2 − β2

x2(m−µx2)

.

The last expression can be factorized into the sum of three factors

Tr(M(µ)) = ϕ(x(y))+ ψ(p)− Tr(Q),

TrQ = ρ(0)

m(0)a(µ),

where the functions ϕ(x(y)) and ψ(p) tend to zero as y resp.p tend to ∞. Thefactorization is unique and obvious

ϕ(x) = x2

m− µx2+ b2(x, µ)

ρ

x2 − β2

x2(m−µx2)

;

ψ(p)= 1

a(µ)

1

p2 + c(µ) +b2(0, µ)

a(µ)

1/4 − c(µ)p2 + c(µ) + ρ(0)

m(0)a(µ).

(49)


Proposition A.1 implies that the essential of the resolvent operator is given by

σess(M(µ)) = (Range(ϕ(x)) ∪ Range(ψ(x))+ ϕ(0)). (50)

Straightforward calculations imply

σess(L) = Rangex∈[0,1]

{m− β2

ρ

x2

}∪[

l0

4 + ρ(0)m(0)

,l0ρ(0)m(0)

], (51)

where l0 is given by (40). The parameter µ disappears eventually as one can expect.This parameter is pure axillary.

We conclude that the essential spectrum of L consists of two parts havingdifferent origin. The so-called regularity spectrum ([30])

Rangex∈[0,1]

{m− β2

ρ

x2

}

is determined by all coefficients of the operator matrix on the whole interval [0, 1].This part of the spectrum coincides with the limit of the essential spectra of thetruncated operators L(ε)

Rangex∈[0,1]

{m− β2

ρ

x2

}=⋃ε>0

σess(L(ε)).

On the contrary the singularity spectrum[l0

4 + ρ(0)m(0)

,l0ρ(0)m(0)

]

is due to the singularity of the operator matrix at the origin and depends on thebehavior of the matrix coefficients at the origin only. This part of the essentialspectrum is absent for all truncated operators L(ε) and cannot be obtained by thelimit procedure ε → 0. This fact explains the name singularity spectrum givenin [30]. The appearance of this interval of the essential spectrum generated bythe singularity was predicted by J. Descloux and G. Geymonat. Note that the endpoint l0/(ρ(0)/m(0)) of the singularity spectrum always belongs to the interval ofregularity spectrum, since

limx→0

m− β2

ρ

x2= l0

ρ(0)m(0)

.

Remark. Let us remind that the essential spectrum has been calculated providedm(0) = 0 and the quasiregularity conditions are satisfied. If m(0) = 0, the qua-siregularity conditions imply that β(0) = 0 and hence m′(0) = 0. No singularity


appears in the coefficients of the matrix L given by (2). Therefore the operator isregular and its essential spectrum equals to

Rangex∈[0,1]

{m− β2

ρ

x2

}(4).

No singularity spectrum appears in this case.There is another way to describe the singularity spectrum using the roots of the

symbol of the asymptotic Hain–Lüst operator, observed first for a different matrixdifferential operator in [30].

LEMMA 9.1. The singularity spectrum[l0

4 + ρ(0)m(0)

,l0ρ(0)m(0)

]

of the operator L coincides with the set of singular points (roots) of the symbol ofthe asymptotic Hain–Lüst operator

. = {µ ∈ R | ∃p ∈ R ∪ {∞} : a(µ)(p2 + c(µ)) = 0}.Proof. The set of singular points of the symbol a(µ)(p2 + c(µ)) coincides with

the set

. = {µ ∈ R | c(µ) � 0}.Formula (40) implies

. ={µ ∈ R | 0 �

l0 − µ ρ(0)m(0)

µ� 4

}

=[

l0

4 + ρ(0)m(0)

,l0ρ(0)m(0)

].

Note that p = ∞ formally corresponds to right endpoint of the last interval. Thelemma is proven. ✷

In our opinion this connection between the singular set of the symbol of theasymptotic Hain–Lüst operator and the singularity spectrum has general character.Studies in this direction will be continued in one of our forthcoming publications.

Remark. We would like to mention that the regularity spectrum

Rangex∈[0,1]

{m− β2

ρ

x2

}under quasiregularity conditions can be calculated using just the symbol of theHain–Lüst operator. Really trivial calculations show that the regularity spectrumcoincides with the set of real µ for which the principle coefficient of the Hain–Lüstoperator degenerates, i.e. equals zero. Roughly speaking this idea has been utilizedby physicists K. Hain and R. Lüst ([16]) (see also [12]).


10. Semiboundedness of the Operator

In many applications to physics semibounded operators play very important role.Semiboundedness of the considered operator is related to the quasiregularity con-ditions.

THEOREM 10.1. Suppose that the real valued functions q, β, ρ,m satisfy thefollowing conditions:

q ∈ L∞[0, 1], β,m, ρ ∈ C2[0, 1], ρ � c0 > 0. (52)

Then the symmetric operator Lmin corresponding to the operator matrix (2) issemibounded if and only if one of the following three conditions is satisfied

(1) (m− β2/ρ)|x=0 > 0,(2) (m− β2/ρ)|x=0 = 0 and (m− β2/ρ)′|x=0 > 0,(3) (m− β2/ρ)|x=0 = 0 and (m− β2/ρ)′|x=0 = 0 (quasiregularity conditions).

COROLLARY 10.1. Under assumptions of Theorem 10.1 the operator Lmin ad-mits self-adjoint extensions. Every such extension L is a semibounded operator ifand only if one of the conditions (1)–(3) is satisfied.

Proof. Since the coefficients of the matrix L are real valued functions, the defi-ciency indices of Lmin are equal. On the other hand the equation for the deficiencyelement is a system of ordinary differential equations. Therefore the set of solutionshas finite dimension. Hence the operator Lmin always has finite equal deficiencyindices and admits self-adjoint extensions. Theorem 10.1 implies that every suchextension is semibounded if and only if one of the three conditions is satisfied(see [3]). ✷

Proof of Theorem 10.1. Without loss of generality one can suppose that q = 0,since the operator corresponding to the matrix

(q 00 0

)is bounded in H and cannot

change the semiboundedness of the whole operator Lmin.The theorem will be proven by estimating the quadratic form of Lmin defined on

the domain C∞0 [0, 1] ⊕ C∞

0 [0, 1] by the following operator matrix

L =

− d

dxρ

d

dx

d

dx

β

x

−βx

d

dx

m

x2

. (53)

The quadratic form of this operator is

〈LminU,U 〉 = 〈ρu′1, u′1〉 −⟨β

xu2, u

′1

⟩−⟨β

xu′1, u2

⟩+⟨m

x2u2, u2

⟩


=⟨

1 − β√ρ

− β√ρ

m

√ρu′1u2

x

,

√ρu′1u2

x

⟩ . (54)

Considering functions with zero second component U = (u1, 0) we concludethat the operator Lmin is not bounded from above, since the quadratic form coin-cides with the quadratic form of the operator −(d/dx)ρd/dx in this case. Thereforethe operator Lmin is semibounded if and only if it is bounded from below. To getthe second necessary condition for the semiboundedness of the operator considerthe set of functions with zero first component U = (0, u2). The quadratic form isthen given by

〈LminU,U 〉 =⟨m

x2u2, u2

⟩.

Hence the operator Lmin is semibounded only if

m(0) > 0 or m(0) = 0 and m′(0) � 0. (55)

Case A. Suppose that the determinant of the matrix

det

1 − β√ρ

− β√ρ

m

= m− β2

ρ

is negative at point zero (and therefore in a neighborhood of this point as well)

m(0)− β(0)2

ρ(0)< 0. (56)

It follows that the matrix has precisely one negative eigenvalue λ(x) < 0 for smallenough values of x. Let us denote by (α, γ ) the corresponding normalized realeigenvector depending continuously on x in a neighborhood of the origin.

Suppose that α(0) = 0. Then the first equation for the eigenvector implies thatβ(0) = 0 and therefore m(0) < 0 due to (56). This contradicts (55) and thereforeα(0) = 0 in a certain neighborhood of the origin due to the continuity of α.

Consider arbitrary real function h ∈ C∞0 [0, 1] such that the derivative of h

is equal to 1 in the interval (1/4, 1/2) and the family of scaled functions hε =εh(x/ε). The corresponding family of vector functions Uε = (hε,

γ

α

√ρxhε ′) is

well-defined for sufficiently small ε. Since

〈LminUε,Uε〉 =

∫ ε

0

λ(x)ρ

α2hε

′2 dx,


and

‖Uε‖2 =∫ ε

0

(hε2 + γ 2ρ

α2x2hε ′2

)dx,

the quotient 〈LU,U 〉/‖ U ‖2 tends to −∞ as ε → 0. Hence the operator Lmin isnot semibounded in this case.

Case B. Suppose that

m(0)− β(0)2

ρ(0)> 0.

The operator L is semibounded in this case. Indeed the quadratic form can bedecomposed as follows

〈LminU,U 〉 =⟨

1 − β(0)√ρ(0)

− β(0)√ρ(0)

m(0)

√ρu′1

u2

x

,

√ρu′1

u2

x

⟩+

+⟨

0β(0)√ρ(0)

− β√ρ

β(0)√ρ(0)

− β√ρ

m−m(0)

√ρu′1

u2

x

,

√ρu′1

u2

x

⟩.

The first term is positive and can be estimated from below by

const(‖u′1‖2 + ‖u2‖2)

due to the assumption. The second term is subordinated to the first one∣∣∣∣∣∣∣∣∣⟨

0β(0)√ρ(0)

− β√ρ

β(0)√ρ(0)

− β√ρ

m−m(0)

√ρu′1

u2

x

,

√ρu′1

u2

x

⟩∣∣∣∣∣∣∣∣∣

� const∫ 1

0

(xu′1

2 + u22

x

)dx

� const ε∫ ε

0

(ρu′1

2 + u22

x2

)dx + const

∫ 1

ε

(u′1

2 + u22

x2

)dx.

The relative bound const ε can be chosen less than 1 and the second term isbounded for any ε > 0.

Case C. Suppose that

m(0)− β(0)2

ρ(0)= 0 and

d

dx

(m− β2

ρ

)|x=0 < 0.


The quadratic form can be decomposed as

〈LminU,U 〉 =⟨(

1 −β/√ρ−β/√ρ β2/ρ

)√ρu′1u2

x

,

√ρu′1u2

x

⟩+

+⟨(

0 0

0 m− β2/ρ

)

√ρu′1u2

x

,

√ρu′1u2

x

⟩.

Consider the vector function

V ε =(∫ x

0

β(t)

ρ(t)hε

′(t) dt, xhε ′(x)

),

where the scalar hε has been introduced investigating Case A. Calculating the thequadratic form

〈LminVε, V ε〉 =

⟨(m− β2

ρ

)hε

′, hε

′⟩

and estimating the norm

‖V ε‖2 =∫ 1

0

[x2hε

′2 +(∫ x

0

β

ρhε

′ dt)2]

dx

�∫ 1

0[x2hε

′2 + const hε2] dx.

Since (m− (β2/ρ))′|x=0 < 0, the quotient

〈LU,U 〉‖U‖2

tends to −∞ as ε → 0. The operator is not semibounded in this case.

Case D. Suppose that(m− β2

ρ

)|x=0 = 0 and

(m− β2

ρ

)′|x=0 � 0. (57)

The operator Lmin is semibounded in this case due to the following estimate⟨(0 0

0 m− β2/ρ

)

√ρu′1u2

x

,

√ρu′1u2

x

⟩

=∫ 1

0

m− β2

ρ

x2|u2|2 dx � const‖u2‖2,


which is valid since the function (m− (β2/ρ))/x2 from below.The Cases A–D cover all the possibilities. The Theorem is proven. ✷

Appendix A. On the Essential Spectrum of the Triple Sum of Operators inBanach Space

The following simple lemma will be used to calculate the essential spectrum of theseparable sum of pseudodifferential operators. It allows one to pass to the limit informula (58) below when the point λ reaches the discrete spectrum.

LEMMA A.1. Let T,Y,P be bounded operators acting in a Banach space X.Suppose that a certain dotted neighborhood of λ = 0 does not belong to thespectrum of the operator T and the point λ = 0 is not in the essential spectrum ofthe operator T. � Suppose in addition that

Y(T − λ)−1P ∈ S∞ (58)

is a compact operator in the dotted neighborhood. Let RT be the parametrix of theoperator T ([13])

RTT =TRT = I. (59)

Then the operator YRTP is compact

YRTP ∈ S∞. (60)

Proof. The following calculations prove the lemma

YRTP := Y(T − λ)−1(T − λ)RTP= Y(T − λ)−1(I − λRT)P= −λY(T − λ)−1RTP= −λYRT(I − λRT)

−1RTP= 0,

(61)

where the second equality from the end is valid for all λ, 0 < |λ| < 1/‖RT‖ notfrom the spectrum of the operator T

(T − λ)R = I − λR ⇒ (T − λ)−1 =R(I − λR)−1.

The lemma is proven. ✷� These conditions imply that the point λ = 0 is a finite type eigenvalue of T ([13]) or does not

belong to the spectrum of T at all. In the last case the proof of the lemma is trivial.


THEOREM A.1. Let M be the operator sum of three bounded operators Q, Y andP acting in a certain Banach space

M = Q + Y + P, (62)

such that the complement in C of the essential spectrum of the operator Q is con-nected. Suppose that the following two operators are compact for any λ from theregular set of Q

P1

Q − λY ∈ S∞; Y1

Q − λP ∈ S∞. (63)

Then the essential spectrum of the operator M can be calculated as follows

σess(M) \ σess(Q) = [σess(Q + Y) ∪ σess(Q + P)] \ σess(Q). (64)

Proof. It has been proven in [13] (Corollary 8.5, page 204) that if the comple-ment in C of the essential spectrum of a certain bounded operator is connected,then any number λ from the spectrum of the operator, but not from the essentialspectrum is a finite type eigenvalue ([13]), i.e. the pole of the resolvent with fi-nite rank Laurent coefficients with negative indices. Lemma A.1 implies that theoperators

YRQ(λ)P, PRQ(λ)Y (65)

are compact operators, where RQ(λ) is one of the parametrix of the operator Q atpoint λ

(Q − λ)RQ(λ) =RQ(λ)(Q − λ) = I. (66)

Then the following equalities can be proven

(Q + Y − λ)RQ(λ)(Q + P − λ) =Q + Y + P − λ;(Q + P − λ)RQ(λ)(Q + Y − λ) =Q + Y + P − λ. (67)

Let us prove the first equality only, since the prove of the second equality is similar.Formulas (65) imply that

(Q + Y − λ)RQ(λ)(Q + P − λ)= (I + YRQ(λ))(Q + P − λ)=Q − λ+ YRQ(λ)(Q − λ)+ P + YRQ(λ)P

=Q + Y + P − λ.We are going to prove now formula (64) for the essential spectra of the operators

M, Q, Y, and P following the idea of [17], where a similar fact has been proven tothe sum of two operators. Let us prove the following inclusion first

σess(M) \ σess(Q) ⊂ [σess(Q + Y) ∪ σess(Q + P)] \ σess(Q). (68)


Suppose that λ does not belong to the essential spectra of the operators Q, Q + Y,and Q + P, then the operators

Q + Y − λ, RQ(λ), Q + P − λare Fredholm operators as a product of three Fredholm operators. Then formulas(67) imply that the operator

Q + Y + P − λis a Fredholm operator. Hence the point λ does not belong to the essential spectrumof the operator Q + Y + P.

In the second step let us prove the inclusion

σess(M) \ σess(Q) ⊃ [σess(Q + Y) ∪ σess(Q + P)] \ σess(Q). (69)

Suppose that λ does not belong to the essential spectra of the operators M and Q,i.e. that the operators M − λ and Q − λ are Fredholm operators. We are going touse Proposition 8.2 from [17] (see also [13]) stating that if the operators A andB are two bounded operators acting in a certain Banach space and the operatorsAB and BA are Fredholm operators, then the operators A and B are also Fredholmoperators. Formulas (67) imply that the operators

RQ(λ)(Q + Y − λ)RQ(λ)(Q + P − λ)and

RQ(λ)(Q + P − λ)RQ(λ)(Q + Y − λ)are Fredholm operators. Then the proposition implies that the operators RQ(λ)(Q+Y − λ) and RQ(λ)(Q + P − λ) are Fredholm operators. It follows from (66) thatthe operators

Q + Y − λ = (Q − λ)RQ(λ)(Q + Y − λ)and

Q + P − λ = (Q − λ)RQ(λ)(Q + P − λ)are Fredholm operators. It follows that λ does not belong to the essential spectra ofthe operators Q + Y and Q + P. Inclusion (69) is proven.

Formulas (68) and (69) imply (64). The Theorem is proven. ✷Remark. It is possible to get read of the condition that the complement in C of

the essential spectrum the operator Q is connected. Then it is necessary to supposethat the operators YRQ(λ)P, PRQ(λ)Y are compact for any λ outside the essentialspectrum of Q. It is possible to construct three operators Q,P,Y satisfying allconditions of the theorem except the connectivity of C \ σess(Q) but not satisfyingformula (64). This counterexample can be prepared using bilateral shift in theHilbert space X = ?2

Z(?2N,H ⊕ H), where H is a certain infinite dimensional

axillary Hilbert space.


PROPOSITION A.1. Let M be any n × n matrix separable pseudodifferentialoperator generated in the Hilbert space L2(R,Cn) by the symbol

M(y, p) = Q+ Y (y)+ P(p), p = 1

i

d

dy, (70)

where Q is a constant diagonalizable matrix with simple spectrum, and the matrixfunctions Y (y) and P(p) are essentially bounded and satisfy the following twoasymptotic conditions

limx→∞Y (y) = 0 lim

p→∞P(p) = 0. (71)

Then the essential spectrum of the operator M is given by

σess(M) = σess(Q + P) ∪ σess(Q + Y). (72)

Proof. The essential spectra of both operators Q + Y and Q + P contain theessential spectrum of Q

σess(Q) = σ (Q),

where σ (Q) is the spectrum of the matrix Q. To prove this fact one can use pertur-bation theory and the fact that the matrices Y (y), y → ∞ and P(p), p → ∞ areasymptotically small ([24]).

Theorem A.1 implies that

σess(Q + Y + P) \ {0} ⊃⋃n

σess(PN(Q + Y)PN) \ {0} ⊃ σ (Q) \ {0},

(using A = Q + Y, B = P ). It follows that

σess(M) \ {0} = (σess (Q + P) ∪ σess (Q + Y)) \ {0}. (73)

We are going to remove the set {0} from the last formula.Applying the same analysis for the operator M − εI we obtain that

σess(M − εI) \ {0} = (σess (Q − εI + P) ∪ σess (Q − εI + Y)) \ {0}. (74)

This implies that

σess(M(y, p)) \ {ε} = (σess(Q + P) ∪ σess(Q + Y)) \ {ε},for arbitrary real ε and hence

σess(M) = (σess(Q + P) ∪ σess(Q + Y)). (75)

The proposition is proven. ✷In the special case case n = 1 and when the symbols Y (y) and P(p) are

piecewise continuous the last proposition can be derived from Theorem 3 in [35](see also [36]). The advantage of our approach is its transparency compared withthe technique of C∗ algebras used in [35].


Remark. The condition concerning the simplicity of the spectrum of matrixQ can be removed in the special case where all matrices Q,Y (y), and P(p) areHermitian.

Proof. Consider the family of small Hermitian perturbations Qε, ε > 0 of thematrix Q such that

‖ Qε −Q ‖� ε

and the spectrum of Qε is simple. Such matrix Qε satisfy the conditions of thetheorem and hence

σess(Qε + Y + P) = σess(Qε + P) ∪ σess(Qε + Y). (76)

Let us denote by Fδ the δ-neighborhood of any set F ⊂ R

Fδ := {x ∈ R : dist(x, F ) � δ}.LetA and B be two bounded self-adjoint operators acting in a certain Hilbert space.Then the essential spectra of the operators A and A+B are related by the followingformula ([3])

σess(A + B) ⊂ (σess(A))‖B‖.

From (76) we immediately obtain that

σess(Q + Y + P(p)) ⊂ [σess(Qε + Y) ∪ σess(Qε + P)]2ε;σess(Q + Y) ∪ σess(Q + P) ⊂ [σess(Qε + Y + P)]2ε.

Since the essential spectra are closed sets and ε is arbitrary small, we conclude that

σess(Qε + Y + P) = σess(Q + Y) ∪ σess(Q + P). (77)

This completes the proof. ✷

Appendix B. Elementary Lemmas on Calkin Calculus

The following lemmas are necessary for the transformation of the resolvent.

LEMMA B.1. Let the real valued function f (y) be positive bounded and sepa-rated from zero

0 < c � f (y) � C (78)

for some c, C ∈ R+. Let the function g(y) be bounded and the operator

L ≡ pf (y)p + g(y) (79)


be self-adjoint and invertible in L2(R). Suppose that the operator pL−1p bebounded. � Then for any bounded function h(y) such that limy→∞ h(y) = 0 thefollowing equality holds in Calkin algebra

pL−1ph = h

f. (80)

Comment 1. The role of the function h is to regularize the equality which doesnot hold in Calkin algebra pL−1 = 1/f . Therefore the regularizing function h can-not be cancelled in (80). To construct a counter example let us first consider similarproblem on the whole axis for which all calculations are trivial. Let the functionsf and g be constant functions f = 1, g = 1. Then the operator

pL−1p − 1 = p(p2 + 1)−1p − 1 = − 1

p2 + 1

obviously is not compact, since it is a multiplication operator in the Fourier repre-sentation.

Comment 2. In [4] similar result has been obtained in the regular case. Herean abstract proof of a generalization of the result is presented. We hope that thealgebraic character of the proof will enable us to generalize these results to a widerclasses of PDO and CDO. The advantage of our approach is that no informationconcerning the Green’s function is used.

Proof. Consider first the case where the function h(y) is a C∞(R) function.We need this condition in order to avoid to consider the closure of bounded op-erators considered below. All these operators are well defined by their differentialexpressions on W 1

2 (R).The following identity holds (at least in W 1

2 (R))

((p + i)L−1(p − i))(

1

p − i L1

p + i)= I.

Multiplying the latter equality by the operator of multiplication by decreasingfunction h one can get the operator equality valid on W 1

2 (R)

((p + i)L−1(p − i)) 1

p − i (pfp)1

p+ i h+

+ ((p + i)L−1(p − i)) 1

p − i g1

p + i h = h.

The second term in left-hand side is a compact operator as the multiplication ofthe bounded operator (p + i)L−1(p − i)(1/(p − i))g and the compact operator

� The latter condition could follow from the previous conditions for sufficiently smoothfunction f .


(1/(p + i))h (since the functions 1/(p + i) and h are decreasing function of pand y respectively). Hence the following equality holds in Calkin algebra

((p + i)L−1(p − i)) 1

p − i (pfp)1

p+ i h =h.

Similarly taking into account that

p1

p + i h =(

1 − i

p + i)h =h

we get the following equality

((p + i)L−1(p − i)) 1

p − i pf h =h.

Multiplying by f −1 the latter equality, one gets

(p + i)L−1(p − i) 1

p − i ph =h

f,

using the fact the function f is boundedly invertible. Multiplying the latter equalityby factor

p

p + i = 1 − i

p + ifrom the left one gets in Calkin algebra

h

f= p

p + i (p + i)L−1(p − i) p

p − i hp

p − i = pL−1ph.

Let us consider the case of decreasing bounded but otherwise arbitrary func-tion h. Every such function can be estimated from above by a certain positivedecreasing to zero C∞(R) function h, |h(y)| � h. We have already proven theLemma for the function h

h

f= p

p + i (p + i)L−1(p − i) p

p − i hp

p − i = pL−1ph.

Of cause the multiplication by the contraction operator of multiplication by thebounded function h/h preserves the equality in Calkin algebra. Finally one gets(80) for arbitrary h satisfying the conditions of the Lemma. ✷

The following lemma is well-known. (It is a special case of problems treatedsystematically in [19].)

LEMMA B.2. Let the following conditions be satisfied


(i) f � c > 0,(ii) f, g ∈ C1(R),

(iii) (pfp + g)|W 22 (R)

is invertible, then the operator (p + i)(pfp + g)−1(p − i)defined originally on the dense set W 1

2 (R) is bounded in L2(R).

COROLLARY. Under conditions of the lemma the operator

p(pfp + g)−1p|W 12 (R)

is also bounded, since the operator p/(p ± i) is a contraction.

Remark. Condition (iv) can be substituted by a stronger condition (iv′) the realvalued function g is positive definite g(x) � c0 > 0. Really conditions (i), (ii), (iii)(iv′) imply (iv), since the estimate

〈(pfp + g)u, u〉 = 〈fpu, pu〉 + 〈gu, u〉 � c0‖pu‖2 + c0‖u‖2

implies that the operator (pfp + g)|W 22 (R)

has bounded inverse.

LEMMA B.3. Suppose that conditions (i)–(iii) of Lemma B.2 be satisfied. Let inaddition the limits

limy→∞f (y), lim

y→∞ g(y),

be finite. Then the difference between the inverse Hain–Lüst and asymptotic Hain–Lüst operators is a compact operator, moreover

(p + i)[T −1(µ)− T −1as (µ)] ∈ S∞;

[T −1(µ)− T −1as (µ)](p − i) ∈ S∞. (81)

Proof. Consider the following chain of equalities

(p + i)[T −1(µ)− T −1as (µ)] ∈ S∞

= (p + i)T −1(µ){Tas(µ)− T (µ)]T −1(µ)

= (p + i)T −1(µ)(pf p + g)T −1(µ)

= (p + i)T −1(µ)(p − i)(p − i)−1(pf p + g)T −1(µ),

where

f = f − limy→∞ f (y) ∈ C

1(R),

g = g − limy→∞ g(y) ∈ L

∞(R) ∩ C1(R),

limy→∞ f (y) = lim

y→∞ g(y) = 0.


The operator (p+ i)T −1(µ)(p− i) is bounded and the operator (p− i)−1(pf p+g)T −1(µ) is compact operators, since the operators

(p − i)−1g and f pT −1(µ)

are compact. (Here we use the fact that any pseudodifferential operator deter-mined by the symbol ϕ(x)ψ(p) is compact if ϕ,ψ ∈ C(R) and limx→∞ ϕ(x) =0, limp→∞ ϕ(p) = 0.) The lemma is proven. ✷LEMMA B.4. Let conditions (i)-(iii) of Lemma B.2 be satisfied. Suppose in ad-dition that the continuous functions α, γ ∈ C(R) are continuous and have finitelimits at infinity. Then the following equality holds in Calkin algebra

(αp + γ )T −1(µ) = (α(∞)p + γ (∞))T −1as (µ). (82)

Proof. Lemma B.3 implies that

(αp + γ )T −1(µ)

= (αp + γ )T −1as (µ)

= (α(∞)p + γ (∞))T −1as (µ)+ ((α − α(∞))p + γ − γ (∞))T −1

as (µ)

= (α(∞)p + γ (∞))T −1as (µ)+ (α − α(∞))pT −1

as (µ)+ (γ − γ (∞))T −1as (µ)

= (α(∞)p + γ (∞))T −1as (µ).

The lemma is proven. ✷Remark. All lemmas proven in this appendix for the operators acting in L2(R)

are in fact valid for the corresponding operators restricted to L2(R+). To make theoperators self-adjoint in L2(R+) one needs to introduce some additional symmetricboundary condition at the origin, for example the Dirichlet boundary condition dis-cussed in the paper. Let us mention here the necessary modifications of Lemma B.1only. The other lemmas can be treated in the same way. Let L be a self-adjoint op-erator in L2(R+) determined by (79) and certain boundary condition at the origin.Consider the extension of L to the operator acting inL2(R) determined by the sameexpression, where the functions f (y) and g(y) are continued for negative valuesof y as even functions. Then equality (80) holds in Calkin algebra for the extendedoperator. Taking into account that the resolvent of the extended operator differsfrom the orthogonal sum of two copies of the resolvents of the initial operator takenon the positive and negative semiaxes separately by a finite rank operator. (Note thatthe functions from the domain of both operators satisfy proper separating boundaryconditions at the origin.) We have used here the Glazman splitting method ([3]).As a result, we obtain the necessary equality for the operators in L2(R+).

Acknowledgements

We would like to thank the referee for valuable remarks which allowed us toimprove the original manuscript.


The work of the authors was supported by the Royal Swedish Academy ofSciences.

References

1. Adam, J. A.: Critical layer singularities and complex eigenvalues in some differential equationsof mathematical physics, Phys. Rep. 142 (1986), 263–356.

2. Adamyan, V., Langer, H., Mennicken, R. and Saurer, J.: Spectral components of self-adjointblock operator matrices with unbounded entries, Math. Nachr. 178 (1996), 43–80.

3. Akhiezer, N. I. and Glazman, I. M.: Theory of Linear Operators in Hilbert Space, Pitman,Boston, 1981.

4. Atkinson, F. V., Langer, H., Mennicken, R. and Shkalikov, A.: The essential spectrum of somematrix operators, Math. Nachr. 167 (1994), 5–20.

5. Descloux, J. and Geymonat, G.: Sur le spectre essentiel d’un operateur relatif á la stabilité d’unplasma en géometrie toroidal, C.R. Acad. Sci. Paris 290 (1980), 795–797.

6. Everitt, W. N. and Bennewitz, C.: Some remarks of the Titchmarsch–Weyl m-coefficient, In:Tribute to Åke Pleijel, Univ. of Uppsala, 1980.

7. Everitt, W. N. and Markus, L.: Boundary Value Problems and Symplectic Algebra for OrdinaryDifferential and Quasi-differential Operators, Amer. Math. Soc., Providence, RI, 1999.

8. Faierman, M., Lifschitz, A., Mennicken, R. and Möller, M.: On the essential spectrum of adifferentially rotating star, ZAMM Z. Angew. Math. Mech. 79 (1999), 739–755.

9. Faierman, M., Mennicken, R. and Möller, M.: The essential spectrum of a system of singularordinary differential operators of mixed order. I. The general problem and an almost regularcase, Math. Nachr. 208 (1999), 101–115.

10. Faierman, M., Mennicken, R. and Möller, M.: The essential spectrum of a system of singularordinary differential operators of mixed order. II. The generalization of Kako’s problem, Math.Nachr. 209 (2000), 55–81.

11. Faierman, M. and Möller, M.: On the essential spectrum of a differentially rotating star in theaxisymmetric case, Proc. Royal Soc. Edinburgh A, 130 (2000), 1–23.

12. Goedbloed, J. P.: Lecture notes on ideal magnetohydrodynamics, In: Rijnhiuzen Report, Form-Instituut voor Plasmafysica, Nieuwegein, 1983, 83–145.

13. Gohberg, I., Goldberg, S. and Kaashoek, M.: Classes of Linear Operators, vol. 1, Birkhäuser,Basel, 1990.

14. Grubb, G.: Functional Calculus of Pseudodifferential Boundary Problems, 2nd edn,Birkhäuser, Basel, 1986.

15. Grubb, G. and Geymonat, G.: The essential spectrum of elliptic systems of mixed order, Math.Anal. 227 (1977), 247–276.

16. Hain, K. and Lüst, R.: Zur Stabilität zylindersymmetrischer Plasmakonfigurationen mitVolumenströmmen, Z. Naturforsch. A 13 (1958), 936–940.

17. Hardt, V., Mennicken, R. and Naboko, S.: System of singular differential operators of mixedorder and applications to 1-dimensional MHD problems, Math. Nachr. 205 (1999), 19–68.

18. Hartman, P.: Ordinary Differential Equations, John Wiley, New York, 1964.19. Hörmander, L.: The Analysis of Linear Partial Differential Operators, vols I–IV, 2nd edn,

Springer-Verlag, Berlin, 1994.20. Kako, T.: On the essential spectrum of MHD plasma in toroidal region, Proc. Japan Acad. Ser.

A Math. Sci. 60 (1984), 53–56.21. Kako, T.: On the absolutely continuous spectrum of MHD plasma confined in the flat torus,

Math. Methods Appl. Sci. 7 (1985), 432–442.22. Kako, T.: Essential spectrum of linearized operator for MHD plasma in cylindrical region,

Z. Angew. Math. Phys. 38 (1987), 433–449.


23. Kako, T. and Descloux, J.: Spectral approximation for the linearized MHD operator incylindrical region, Japan J. Indust. Appl. Math. 8 (1991), 221–244.

24. Kato, T.: Perturbation Theory for Linear Operators, 2nd edn, Springer, Berlin, 1976.25. Konstantinov, A. and Mennicken, R.: On the Friedrichs extension of some block operator

matrices, Integral Equations Operator Theory 42 (2002), 472–481.26. Krasnosel’skii, M. A., Zabreiko, P. P., Pustylnik, E. I. and Sobolevskii, P. E.: Integral Operators

in Spaces of Summable Functions, Noordhoff Publishing, Leiden, 1976.27. Kurasov, P.: On the Coulomb potential in one dimension, J. Phys. A 29 (1996), 1767–1771;

Response to ‘Comment on “On the Coulomb potential in one dimension”,’ J. Phys. A 30 (1997),5579–5581.

28. Langer, H. and Möller, M.: The essential spectrum of a non-elliptic boundary value problem,Math. Nachr. 178 (1996), 233–248.

29. Lifchitz, A. E.: Magnetohydrodynamics and Spectral Theory, Kluwer Acad. Publ., Dordrecht,1989.

30. Mennicken, R., Naboko, S. and Tretter, Ch.: Essential spectrum of a system of singulardifferential operators, to appear in Proc. Amer. Math. Soc.

31. Möller, M.: On the essential spectrum of a class of operators in Hilbert space, Math. Nachr. 194(1998), 185–196.

32. Naboko, S.: Nontangential boundary values of operator-valued R-functions in a half-plane,St. Petersburg (Leningrad) Math. J. 1 (1990), 1255–1278.

33. Naimark, M. A.: Linear Differential Operators, Ungar, New York, 1968.34. Olver, F. W.: Introduction to Asymptotic and Special Functions, Academic Press, New York,

1974.35. Power, S. C.: Essential spectra of piecewise continuous Fourier integral operators, Proc. Roy.

Irish. Acad. 81 (1981), 1–7.36. Poer, S. C.: Fredholm theory of piecewise continuous Fourier integral operators on Hilbert

space, J. Operator Theory 7 (1982), 51–60.37. Raikov, G. D.: The spectrum of a linear magnetohydrodynamic model with cylindrical

symmetry, Arch. Rational Mech. Anal. 116 (1991), 161–198.38. Rofe-Beketov, F.: Self-adjoint extensions of differentila operators in a space of vector-

functions, Soviet. Math. Doklady 10 (1969), 188–192.39. Rofe-Beketov, F. and Kholkin, A.: Spectral Analysis of Differential Operators. Relation of the

Spectral and Oscillatory Properties, Mariupol, 2001.40. Schechter, M.: On the essential spectrum of an arbitrary operator. I, J. Math. Anal. Appl. 13

(1966), 205–215.41. Schechter, M.: Spectra of Partial Differential Operators, 2nd edn, North-Holland, Amsterdam,

1986.42. Reed, M. and Simon, B.: Methods of Modern Mathematical Physics, 2nd edn, vols I–IV.

Academic Press, New York, 1984.43. de Snoo, H.: Regular Sturm–Liouville problems whose coefficients depend rationally on the

eigenvalue parameter, Math. Nachr. 182 (1996), 99–126.44. Zelenko, L. B. and Rofe-Beketov, F. S.: The limit spectrum of systems of first order differential

equations with slowly changing coefficients (Russian), Differencial’nye Uravnenija 7 (1971),1982–1991 (in Russian).


287

A Construction of Berezin–Toeplitz Operatorsvia Schrödinger Operators and the ProbabilisticRepresentation of Berezin–Toeplitz SemigroupsBased on Planar Brownian Motion

BERNHARD G. BODMANNDepartment of Physics, Princeton University, 337 Jadwin Hall, Princeton, NJ 08544, U.S.A.e-mail: [email protected]

(Received: 12 December 2001)

Abstract. First we discuss the construction of self-adjoint Berezin–Toeplitz operators on weightedBergman spaces via semibounded quadratic forms. To ensure semiboundedness, regularity con-ditions on the real-valued functions serving as symbols of these Berezin–Toeplitz operators areimposed. Then a probabilistic expression of the sesqui-analytic integral kernel for the associatedsemigroups is derived. All results are the consequence of a relation of Berezin–Toeplitz operators toSchrödinger operators defined via certain quadratic forms. The probabilistic expression is derived inconjunction with the Feynman–Kac–Itô formula.

Mathematics Subject Classifications (2000): 81S10, 47D08.

Key words: weighted Bergman spaces, Berezin–Toeplitz operators, Schrödinger operators, semi-groups, Feynman–Kac–Itô formula.

1. Introduction

The results presented here are inspired by a concept of Daubechies and Klauder[14, 15] which provides a probabilistic expression for the unitary group generatedby Hamiltonians arising from the so-called anti-Wick quantization prescription.Based on geometric considerations, several works [1, 2, 26, 27] advocate naturalgeneralizations of this expression which can be related [25, 29] to Hamiltoniansfrom the more universal Berezin–Toeplitz quantization scheme [6].

The generalization presented in this paper evolved from a pattern behind theconstruction in [14, 15], according to which certain Berezin–Toeplitz operatorscan be realized as limits of monotone families of Schrödinger operators. Herebya Berezin–Toeplitz operator Tf is understood as the compression of a suitablemultiplication operator Mf : ψ �→ fψ from a Hilbert space of square-integrablefunctions to a closed subspace of analytic functions. In this context the real-valuedfunction f is called a symbol and can be interpreted as a classical observablecorresponding to the operator Tf . In quantum mechanics Berezin–Toeplitz oper-

288 BERNHARD G. BODMANN

ators model a variety of systems with canonical or other degrees of freedom [5, 6].A Schrödinger operator H , on the other hand, can be thought of as a second-orderdifferential operator on a subset � of d-dimensional Euclidean space R

d , formallywritten H = ∑d

k=1(i∂k + ak)2 + υ with the vector potential a: � → Rd and the

scalar potential υ: � → R. Fairly general conditions have been worked out underwhich this formal expression characterizes H uniquely as a self-adjoint operator[11, 13, 20, 21, 28].

The first task in this paper is to find conditions on f which guarantee that Tfcan be defined as a self-adjoint operator. Secondly we derive the analogue of theprobabilistic expression in [14] for Berezin–Toeplitz semigroups in this generalsetting. Unlike the results [14, 16] derived in analogy of the anti-Wick situation andthe strategies advocated for their generalization [1, 2, 26, 27], we stay with a proba-bilistic representation which is based on standard Brownian motion [36]. This wayone may benefit from the associated repertory of probabilistic techniques. A minordifference with the original [14] is that we are concerned with Berezin–Toeplitzsemigroups, not with the unitary groups associated with quantum mechanical timeevolution.

The structure and contents of this paper are as follows:After fixing the notation, we start in Section 3 with a review of Hilbert spaces

of analytic functions which are known as weighted Bergman spaces. In Sections 4and 5 we construct Berezin–Toeplitz operators on these spaces and state sufficientconditions on their symbols which guarantee self-adjointness and semibounded-ness. Hereby an essential tool is that certain Berezin–Toeplitz operators can beextended to self-adjoint Schrödinger operators which are defined via quadraticforms.

In Section 6 we continue the spirit of [14, 15] and derive a probabilistic ex-pression for Berezin–Toeplitz semigroups which uses a realization of Berezin–Toeplitz operators as limits of monotone families of Schrödinger operators andthe probabilistic representation of Schrödinger semigroups with the help of theFeynman–Kac–Itô formula [11, 38, 40].

This paper is not entirely self-contained. For the relevant background informa-tion, the reader is referred to [37] and [11] which neatly comprise the essentialbuilding blocks for the results presented here.

2. Basic Definitions

Let R denote the real numbers. By convention, D is always an open, simply con-nected set in the plane of complex numbers C := {z = z1 + iz2 : z1,2 ∈ R}. Theboundary of a set A ⊂ C is written as ∂A, its complement in C as C \ A or Ac.We will denote the real and imaginary parts of a complex number z as z1 and z2,respectively, and the complex conjugate as z = z1 − iz2.

All functions appearing in this text are tacitly understood to be measurable, eachone in its appropriate sense. The positive and negative parts f + and f − of a real-

A CONSTRUCTION OF BEREZIN–TOEPLITZ OPERATORS 289

valued function f are defined by f ± := max{±f, 0}, such that f = f + −f −. Theindicator function of a set A is denoted as χA.

Several spaces of complex-valued functions are used in the text. The space ofarbitrarily often differentiable functions with compact support inside D is referredto as C∞

c (D). Concerning differentiability, D is hereby regarded as a subset of thereal vector space C ∼= R

2. The space of Lebesgue-essentially bounded functionson D is denoted as L∞(D).

DEFINITION 2.1. A complex-valued function φ: D → C defined on D is calledanalytic in D if it is differentiable and satisfies the Cauchy–Riemann differentialequations, which are stated as

(∂1 + i∂2)φ(z1 + iz2) = 0, (1)

with the partial derivatives ∂1,2 := ∂/∂z1,2.

DEFINITION 2.2. A positive function g: D →]0,∞[ is said to be essentiallybounded away from zero on compacts inside D if for any given compact set C ⊂ D

there exists a δ > 0 such that

ess infz∈C

g(z) := supA

infz∈A g(z) � δ. (2)

Hereby the supremum is taken over all Lebesgue-measurable subsets A ⊂ C suchthat the Lebesgue measure λ vanishes on their difference, λ(C \A) = 0.

DEFINITION 2.3. A real-valued function f : C → R belongs to the Kato classKin two dimensions [24, 40] if the following condition is satisfied:

limr↘0

supz∈C

∫{|y−z|<r}

|f (y)|ln|z− y| dλ(y) = 0. (3)

Whenever this property only holds locally, that is, χCf ∈ K for all compact sets Cin C, we write f ∈ Kloc. It is useful to know that the local Kato property implieslocal integrability with respect to the Lebesgue measure, Kloc ⊂ L1

loc(C).If a function satisfies f + ∈ Kloc and f− ∈ K then it is called Kato decompos-

able, symbolized as f ∈ K±.In order to apply these notions to functions which are at first only defined on a

subset D ⊂ C, they are by convention extended to be zero on Dc.

3. Hilbert Spaces of Analytic Functions

DEFINITION 3.1. Given a positive function g: D →]0,∞[ we associate with itthe so-called weighted Bergman space

L2a(gλ) := {φ: D → C, analytic in D and (φ, φ) <∞}, (4)


a vector space which is endowed with the inner product

(φ,ψ) :=∫

D

φ(z)ψ(z)g(z) dλ(z) (5)

where λ stands for the Lebesgue measure on the complex plane.

Remarks 3.2. As a special case, if D is a bounded domain and the weightfunction is constant, L2

a(gλ) is the well-known Bergman space [7, 30].The inner product suggests that L2

a(gλ) can be identified with a vector-subspaceof L2(gλ). To be precise, we recall that the latter consists of equivalence classesof gλ-square-integrable functions on D which differ from each other on a set ofLebesgue measure zero. Consequently, we identify each function in L2

a(gλ) withits equivalence class from L2(gλ).

Now we will state conditions which guarantee that L2a(gλ) forms a Hilbert-

subspace of L2(gλ).

PROPOSITION 3.3. If g is essentially bounded away from zero on compacts in-side D, then L2

a(gλ) is complete with respect to the norm-topology induced by theinner product.

Proof. Let (ψn)n∈N be a Cauchy sequence in L2a(gλ). First we show uniform

convergence of ψn on compacts inside D. Consider a compact subset C ⊂ D anda safety radius r < infy∈∂D |z − y| for all z ∈ C. By assumption there is a lowerbound δ > 0 such that (2) is satisfied. Using the mean value property for ana-lytic functions, Jensen’s inequality in conjunction with the convex square-modulusfunction, and the lower bound for g we estimate

supz′∈C

|ψn(z′)− ψm(z′)|2

= supz′∈C

∣∣∣∣ 1

πr2

∫B(r,z′)

(ψn(z)− ψm(z)) dλ(z)

∣∣∣∣2

(6)

� 1

πr2supz′∈C

∫B(r,z′)

|ψn(z)− ψm(z)|2 dλ(z) (7)

� 1

πr2δ

∫B(r,z′)

|ψn(z)− ψm(z)|2 g(z) dλ(z) (8)

� 1

πr2δ‖ψn − ψm‖2. (9)

The right-hand side can be made arbitrarily small and thus the sequence (ψn) con-verges uniformly on C. We can therefore conclude that the pointwise limit definesa function ψ : ψ(z) = limn→∞ ψn(z) which is also analytic in D.


It remains to show that the convergence ψn → ψ is also in the sense of thenorm. Due to pointwise convergence and Fatou’s lemma the inequality

‖ψ − ψm‖ � lim infn→∞ ‖ψn − ψm‖ (10)

follows, therefore the Cauchy property entails norm convergence. ✷LEMMA 3.4. Under the same assumption on g as in the preceding proposition,the point-evaluation functionals Fz parameterized by z ∈ D which are definedaccording to

L2a(gλ) −→ C, (11)

Fz:ψ �−→ ψ(z) (12)

are continuous linear mappings.Proof. Let C be a compact neighborhood of z and select a convergent sequence

(ψn)n∈N in L2a(gλ). Since the sequence has the Cauchy property, we can use the

chain of inequalities (6) to show that ψn(z) is also Cauchy, and therefore conver-gent. ✷PROPOSITION 3.5. If g is essentially bounded away from zero on compacts in-side D, the weighted Bergman space L2

a(gλ) possesses a reproducing kernel. Moreexplicitly, there is a kernel κ: D × D → C such that any function ψ in L2

a(gλ)

satisfies the integral equation

ψ(z′) =∫

D

κ(z′, z)ψ(z) g(z) dλ(z). (13)

Proof. To see this, we observe that due to the continuity of the functional Fz andthe completeness of L2

a(gλ) the Riesz representation theorem implies that there isa vector κz in L2

a(gλ) such that

(κz, ψ) = ψ(z) for all ψ ∈ L2a(gλ). (14)

Inserting the definition of the inner product (5) yields the desired integral Equa-tion (13) with the claimed kernel given by κ(z, z′) = (κz, κz′). ✷

The last proposition ensures that all bounded operators have integral kernels.

COROLLARY 3.6. If g is essentially bounded away from zero on compacts in-side D, then any bounded operator B on L2

a(gλ) possesses an integral kernel givenby B(z, z′) = (κz, Bκz′), which means that the image of ψ ∈ L2

a(gλ) is expressedas

Bψ(z) =∫

D

B(z, z′)ψ(z′) g(z′) dλ(z′). (15)


Proof. That B(z, z′) is indeed an integral kernel results from the reproducingproperty (13) and Fubini’s theorem. The sesqui-analyticity follows from the point-evaluation property (14) and the analyticity of the functions in L2

a(gλ). ✷Remark 3.7. Since (15) makes sense even for ψ ∈ L2(gλ), any bounded oper-

ator extends naturally via its integral kernel to L2(gλ). From this point of view, κis the integral kernel of an orthogonal projection operator, henceforth called K ,which maps L2(gλ) onto L2

a(gλ).

EXAMPLES 3.8. Various examples of Lie group representations are realized onspecific weighted Bergman spaces. We list a few groups and their unitary irre-ducible action which is in all but the last example related to Moebius transforma-tions on the associated representation spaces. For more details, see [32] or [31].Unless otherwise stated we set D = C.

(1) Heisenberg–Weyl group. Hereby the Hilbert space is specified by the weightfunction g(z) = (1/π)e−|z|2 . The reproducing kernel is κ(z, z′) = ezz

′. This

space is also known as Fock–Bargmann space [3]. The group representationD(α, β) with parameters α ∈ [0, 2π [, β ∈ C acts on a vector ψ by

D(α, β)ψ(z) = eiαe−|β|2 eβzψ(z − β). (16)

(2) SU(2) group. For each integer or half-integer j ∈ 12N a (2j + 1)-dimensional

space is defined by setting

g(z) = 2j + 1

π(1 + |z|2)−2j−2.

The reproducing kernel is κ(z, z′) = (1 + zz′)2j . The group parameterized byα, β ∈ C with |α|2 + |β|2 = 1 acts as

D(α, β)ψ(z) = (βz + α)2jψ(αz− ββz + α

). (17)

(3) SU(1, 1) group. There are two well-known ways to represent this group onweighted Bergman spaces.(a) The first one is described in [32]. Unlike the previous examples, here D is

not the whole complex plane, but the unit disc D = {z: |z| < 1}, and

g(z) = 2k − 1

π(1 − zz)2k−2

with a fixed number k ∈ {1, 3/2, 2, . . .}. The reproducing kernel isκ(z, z′) = (1 − zz′)−2k. The group action is given with the parametersα, β ∈ C, |α|2 − |β|2 = 1 as

D(α, β)ψ(z) = (βz + α)−2kψ

(αz+ ββz + α

). (18)


(b) An alternative to [32] comes from the so-called Barut–Girardello repre-sentation [4] of the SU(1, 1) group. We set D = C again, select k as aboveand choose

g(z) = 2|z|2k−1

π+(2k)K2k−1(2|z|)

with the gamma function + and the modified Bessel function

Kσ (r) = 1

2(r/2)σ

∫ ∞

0t−σ−1 exp(−t − r2/4t) dt

for r > 0, σ ∈ R [19, 8.432(6)]. The kernel is given as the confluenthypergeometric limit function

κ(z, z′) = 0F1(2k, zz′) =

∞∑n=0

(zz′)n

(2k)n+(n+ 1).

Hereby (2k)n denotes the Pochhammer symbol, defined by (2k)0 := 1and the recursion (2k)n = (2k)n−1(2k + n − 1). The group action, how-ever, does not seem to be related to Moebius transformations and will beomitted here.

Remarks 3.9. In all these examples the kernel can be constructed from theweight function with the help of the Gram–Schmidt orthogonalization procedure.Due to the rotational symmetry g(z) = g(|z|), monomials pn: z �→ zn with differ-ing degree n ∈ {0, 1, 2, . . .} are orthogonal, so the kernel is diagonalized in termsof the basis functions as κ(z, z′) = ∑

n cnpn(z)pn(z′) with suitable normalization

constants cn. Here the summation runs for all the examples over the nonnegativeintegers, except for the SU(2) case, because there, only monomials with maximaldegree 2j are square integrable.

In Section 6 we will present a probabilistic approach to construct the reproduc-ing kernel, which does not rely on special symmetries of g.

4. Self-adjoint Berezin–Toeplitz Operators Defined by Quadratic Forms

In the remaining text we assume that L2a(gλ) is complete.

DEFINITION 4.1. Given the Hilbert space L2a(gλ) and a real-valued function

f : D → R, we consider the sesquilinear form

Q(tf )× Q(tf ) −→ C, (19)

tf :(ψ, φ) �−→

∫D

fψφg dλ, (20)


with form domain

Q(tf ) :={ψ ∈ L2

a(gλ) :∫

D

|fψ2|g dλ <∞}. (21)

When it is interpreted as quadratic form, tf is written as tf (ψ) := tf (ψ,ψ).

The next concern is a condition to guarantee that tf is closed and semibounded,which in turn ensures that there is a self-adjoint operator associated with tf via theFriedrichs representation theorem.

LEMMA 4.2. The sesquilinear form belonging to a nonnegative function f � 0is closed.

Proof. We need to show that Q(tf ), equipped with the form-norm ‖•‖tf definedby

‖ψ‖tf := (tf (ψ)+ ‖ψ‖2)1/2 for ψ ∈ Q(tf ), (22)

is complete.Suppose (ψn)n∈N is a Cauchy sequence with respect to the form-norm. Due

to the estimate ‖ψ‖ � ‖ψ‖tf the sequence is convergent in L2a(gλ), ψn → ψ .

Using pointwise convergence and Fatou’s lemma, we obtain ‖ψ − ψn‖tf �lim infm→∞ ‖ψm − ψn‖tf and therefore the sequence (ψn) converges with respectto the form-norm. ✷THEOREM 4.3. If the form tf+ belonging to the positive part f + � 0 of a func-tion f = f + − f − is densely defined and the negative part can be incorporated intf as a form-bounded perturbation

tf−(ψ) � c1 tf+(ψ)+ c2‖ψ‖2 (23)

with relative form bound c1 < 1 and a constant c2 � 0, then tf is closed onQ(tf ) = Q(tf+) and has a (greatest) lower bound c ∈ R, such that tf (ψ) �c‖ψ‖2.

Proof. This is the so-called KLMN theorem. For the proof, see [33, Theo-rem X.17]. ✷THEOREM 4.4. If the form tf is closed and has the greatest lower bound c as inthe preceding theorem, then it belongs to a unique self-adjoint operator Tf which ischaracterized in terms of the square-root

√Tf − c by the domain D(

√Tf − c) =

Q(tf ) and the equality

(√Tf − c φ,√Tf − c ψ)+ c(φ,ψ)= tf (φ,ψ) for all φ,ψ ∈ Q(tf ). (24)

Proof. Again, we refer to the literature [34, Theorem VIII.15] or [41, Theo-rem 5.36], where this result is known as the Friedrichs representation theorem. ✷


Remarks 4.5. In the context of weighted Bergman spaces we call Tf a self-adjoint Berezin–Toeplitz operator and the function f its symbol.

For ψ ∈ Dmin := {ψ ∈ L2a(gλ), fψ ∈ L2(gλ)} the identity Tfψ = K(f ψ)

relates Tf to the traditional way of defining a Berezin–Toeplitz operator as a com-position of a multiplication operator with the orthogonal projection K . In thatscheme Tf would arise as so-called Friedrichs extension from the quadratic formof the traditionally defined semibounded operator.

A disadvantage of defining Tf by a semibounded form is that in general noth-ing is known about its domain. The situation is different, however, if Dmin is adomain of essential self-adjointness for Tf . Such situations have been investigatedin detail [12, 23] for the case of the Fock–Bargmann space (see Example 1 inSection 3).

5. Relation to Schrödinger Operators

This section shows how a Berezin–Toeplitz operator can be extended to a familyof Schrödinger operators. A major benefit is that the knowledge about Schrödingeroperators can be used to find sufficient conditions for the semiboundedness of tf

in terms of g and f . These ensure self-adjointness of the corresponding Berezin–Toeplitz operator.

5.1. EMBEDDING THE WEIGHTED BERGMAN SPACE

Consider the unitary mapping

L2(gλ) −→ L2(D), (25)

U :ψ �−→ √

gψ, (26)

which simply amounts to a redistribution of the weight function. This mappingidentifies L2

a(gλ) with a closed subspace of L2(D).

Remark. Note that the unitary equivalent K = UKU † of K is an orthogonalprojection operator which maps L2(D) onto U(L2

a(gλ)) and has the integral kernelκ(z, z′) = κ(z, z′)

√g(z)g(z′). However, unlike the case for L2

a(gλ), this kernelis continuous if and only if g is continuous. Following the previous notation wedefine κz′ := √

g(z′) Uκz′.

5.2. A QUADRATIC FORM AND ITS NULL SPACE

The purpose of this subsection is to construct a Schrödinger operator which isnonnegative and has U(L2

a(gλ)) as its null eigenspace.


DEFINITION 5.1. We define the quadratic form

C∞c (D) −→ R, (27)

r:ψ �−→ ‖(i∂1 − ∂2 + a)ψ‖2, (28)

with the function

a(z) := (−i∂1 + ∂2) ln√g(z), (29)

where g is such that the local integrability condition |a|2 ∈ L1loc(D) holds. This

form is closeable and the resulting form-closure domain Q(r) is contained in theset L2(D). We say that the corresponding self-adjoint operator R obeys Dirichletboundary conditions.

Remarks 5.2. Despite the construction of R via (28), in general the null-spaceN (R) := R−1({0}) is only a closed subspace of U(L2

a(gλ)). On the other hand,if instead of C∞

c (D) we start with the maximal form domain, which means theset of all ψ ∈ L2(D) for which the expression on the right-hand side of (28)makes sense and is finite, then the resulting self-adjoint operator is said to obeyNeumann boundary conditions and possesses the null-space U(L2

a(gλ)). For thisreason it seems like an artificial complication to introduce Dirichlet boundary con-ditions here. Nevertheless, the preceding definition is needed as a preparation forthe probabilistic expression in Section 6.

If a is absolutely continuous we can formally write R as a Schrödinger-typeoperator

R = (i∂1 + a1)2 + (i∂2 + a2)

2 + υ (30)

with υ := ∂2a1 − ∂1a2. However, in order to consider R as the usual form sum [11,Remark 2.7] we need additional regularity assumptions on a and υ; see also thefollowing subsection.

Next we establish conditions which guarantee the inclusion of U(L2a(gλ)) in

the domain of R.

PROPOSITION 5.3. If either D = C or the operator Tf specified by f (z) =infy∈∂D |z− y|−2 is bounded on L2

a(gλ), then the image U(L2a(gλ)) is contained in

the form-closure domain Q(r) and the identity N (R) = U(L2a(gλ)) follows.

Proof. In case D = C the argument follows a standard procedure, comparewith [39] or [13, Theorem 1.13]. The more general case treated here demands anadaptation.

The proof proceeds in two steps: First we will show that for any function ψ ∈U(L2

a(gλ)), the space of compactly supported, in D essentially bounded functionsL∞c (D), provides a sequence which converges to ψ in the sense of the form norm.


The second step is a standard mollifier argument, which shows that C∞c (D) is dense

in L∞c (D) ∩ Q(r).

Step 1: Consider a function ψ ∈ U(L2a(gλ)). We perform two alterations: a

mollified cutoff near ∂D (if Dc is nonempty) and a smooth cutoff towards infinity.

The details are as follows: LetDn := {z ∈ D : |z− y| � 1/n for all y ∈ ∂D}.Obviously the sequence of characteristic functions χDn converges pointwise to oneon D as n → ∞. In addition, consider a real-valued function η ∈ C∞

0 (C) whichis nonnegative and satisfies η(0) = maxz∈C η(z) = 1 as well as the gradientbound |∇η| � 1. We define a sequence ηn(z) = η(z/n) which also tends toone from below. Furthermore, let the nonnegative function δ1 ∈ C∞

c (C) be anapproximate δ-function, which means

∫Cδ1(z) dλ = 1, and δ1 = 0 for all |z| > 1.

We define a sequence of approximations δn(z) := n2δ1(nz) to smear out χDn byconvolution,

(χDn ∗ δn)(z) :=∫

D

χDn(y)δn(x − y) dλ(y).

Now consider φn := ψηn(χDn ∗δn). Clearly φn → ψ inL2(D). To show that theconvergence is also with respect to the form-norm, we use the triangle inequality,

‖(i∂1 − ∂2 + a)φn‖= ‖ψ(i∂1 − ∂2)ηn(χDn ∗ δn)‖� ‖ψ(χDn ∗ δn)(i∂1 − ∂2)ηn‖ + ‖ψηn(i∂1 − ∂2)(χDn ∗ δn)‖. (31)

The first term on the right-hand side vanishes in the limit n → ∞ by dominatedconvergence; the second term needs a closer look. Note that the neigborhood of theboundary contains the support

supp(i∂1 − ∂2)(δn ∗ χDn) ⊂ {z ∈ D : |z − y| > 2/n ∀y ∈ ∂D}.By Hölder’s inequality we can estimate

|∇δn ∗ χDn | � nλ(supp δ1)max |∇δ1|.In consequence there is some constant c � 0 such that |(i∂1 − ∂2)(χDn ∗ δn)| �c√f with the function f given in the statement of the theorem. Hence by the

assumption on the boundedness of Tf we can apply dominated convergence andthe second term in (31) also vanishes as n → ∞.

Step 2: Let ψ ∈ L∞c (D) ∩ Q(r) be given. We borrow an argument from [11,

Lemma B.3, Assertion 3)]. Consider the previously defined δn for only suitablylarge n such that ψn := ψ ∗ δn ∈ C∞

c (D).We have limn→∞ ‖φ ∗ δn − φ‖ = 0 for all φ ∈ L2(D). Therefore it remains to

show that after the estimate

‖(i∂1 − ∂2 + a)(ψn − ψ)‖ � ‖(i∂1 − ∂2)(ψn − ψ)‖ + ‖a(ψn − ψ)‖ (32)

each term on the right-hand side converges to zero.


The first term is taken care of by the identity

‖(i∂1 − ∂2)(ψn − ψ)‖ = ‖((i∂1 − ∂2)ψ) ∗ δn − (i∂1 − ∂2)ψ‖ (33)

because the assumption |a|2 ∈ Kloc implies a ∈ L2loc(D) and

‖(i∂1 − ∂2)ψ‖ � ‖(i∂1 − ∂2 + a)ψ‖ + ‖aψ‖ <∞. (34)

To ensure that the second term in (32) vanishes we pass to a subsequence of ψnwhich is almost everywhere convergent and select a compact set C which containsthe support of all but finitely many ψn. The bound |ψn(x)| � χC‖ψ‖∞ for x ∈ D

together with a ∈ L2loc(D) allows dominated convergence which completes the

proof. ✷Remark 5.4. When D is a proper subset of the complex numbers, the condition

on the functions in U(L2a(gλ)) amounts to controlling their decay towards the

boundary ∂D, which relates to the intuitive understanding of Dirichlet boundaryconditions.

5.3. EXTENDING BEREZIN–TOEPLITZ OPERATORS TO SCHRÖDINGER

OPERATORS

Henceforth, we say f and g are admissible if they are such that |a|2 ∈ Kloc and∂2a1 − ∂1a2, f ∈ K±.

For such f and g we define a family {s(ν)f }ν>0 of quadratic forms

Q(s(ν)f ) −→ R, (35)

s(ν)f :

ψ �−→ νr(ψ)+∫

D

f |ψ |2 dλ. (36)

The domain is independent of ν given by

Q(s(ν)f ) := Q(r) ∩ {ψ : ‖√f +ψ‖ <∞},because the negative part of the second term in (36) is, due to the assumption,infinitesimally form-bounded with respect to r.

With the above regularity assumptions on a, υ = ∂2a1 − ∂1a2 and f the corre-sponding self-adjoint, semibounded operator S(ν)f is a Schrödinger operator definedin the usual form sense [11], which is implicitly understood in the expression

S(ν)f = ν[(i∂1 + a1)

2 + (i∂2 + a2)2 + υ] + f. (37)


THEOREM 5.5. If f and g obey the conditions

|a|2 ∈ Kloc and ∂2a1 − ∂1a2, f ∈ K±,

and the conclusion of Proposition 5.3 that N (R) = U(L2a(gλ)) holds, then the

form tf is semibounded and closed on Q(tf+). Therefore, f defines a self-adjointsemibounded operator Tf on the closure Q(tf ) ⊂ L2

a(gλ).Proof. First we note s

(ν)f (Uψ) = tf (ψ) for any ν > 0 and ψ ∈ Q(tf ). Thus,

we only need to show that the restriction of s(ν)f to the closed subspace N (r) =

U(L2a(gλ)) is again a closed and semibounded form.

To show closedness, assume a sequence (ψn)n∈N in N (r) which is Cauchy withrespect to the form-norm. Then by the closedness of s

(ν)f the sequence has a limit

ψ ∈ Q(s(ν)f ). However, this limit must also be contained in N (r), because N (r)is a closed subspace and the sequence (ψn)n∈N converges with respect to the usualnorm on L2(D).

Semiboundedness follows from the inequality

inf{s(ν)f (ψ) : ‖ψ‖ = 1}� inf{s(ν)f (ψ) : ψ ∈ N (r) and ‖ψ‖ = 1}. ✷ (38)

Remarks 5.6. As stated, the above theorem does not imply that tf is denselydefined. Therefore, Tf might be self-adjoint only on a Hilbert-subspace of L2

a(gλ).In analogy with Theorem 4.3, it is sufficient for the closedness and semibound-

edness of tf when for some ν > 0 the negative part f − can be incorporated as aform-bounded perturbation of s

(ν)

f+ with relative form bound strictly less than one.However, this condition is not as easy to characterize in terms of f as the strongerassumption in the preceding theorem.

6. Probabilistic Representation of Berezin–Toeplitz Semigroups

In this section we derive a probabilistic expression for the sesqui-analytic integralkernel of Berezin–Toeplitz semigroups. The major steps are the reconstructionof Tf via the monotone convergence of s

(ν)f for ν → ∞ and the application of

the Feynman–Kac–Itô formula.

DEFINITION 6.1. For a given z, z′ ∈ C and t, ν > 0 we define the integral withrespect to the pinned Wiener measure∫

(•) dµ(ν)z,0;z′,t := 1

4πtνe−|z−z′|2/4tν

E[•] (39)

via the expectation E[•] with respect to the two-dimensional Brownian bridgemeasure with diffusion constant ν. Both measures are concentrated on the set of


continuous paths {b: [0, t] → C, b(0) = z and b(t) = z′} which are pinned at thestart and endpoint. As Gaussian stochastic process the Brownian bridge is uniquelydetermined by its mean

E[b(s)] = z+ (z′ − z)st, s ∈ [0, t] (40)

and covariance

E[b(r)b(s)] − E[b(r)]E[b(s)] = 4ν

(min{r, s} − rs

t

), (41)

E[b(r)b(s)] − E[b(r)]E[b(s)] = 0 r, s ∈ [0, t]. (42)

DEFINITION 6.2. Given D, the random variable TD := inf{s > 0 : b(s) ∈ Dc} is

called the first exit time of the process. By convention, we define TD to be infiniteon the set for which b never leaves D.

PROPOSITION 6.3. If f and g are admissible and N (R) = U(L2a(gλ)) as in the

conclusion of Proposition 5.3, then for ν → ∞, the semigroup generated by S(ν)fconverges strongly,

limν→∞ e−tS(ν)f ψ = e−tS(∞)f Eψ = Ue−tTf E U †ψ, (43)

where ψ ∈ L2(D), E = E†E projects onto the closure Q(tf ), and E := UEU †.Proof. The limit ν → ∞ of s

(ν)f yields a nondensely defined form

s(∞)f : ψ �→ lim

ν→∞ s(ν)f (ψ) (44)

with the domain Q(s(∞)f ) = U(Q(tf )). This last equality follows from Proposi-tion 5.3 and the definition of the embedding.

The monotone convergence implies that s(∞)f is closed [37] and semibounded-

ness follows from that of s(ν)f for any ν > 0. All these properties hold then for tf

as well, via the identity s(∞)f (Uψ) = tf (ψ) valid for all ψ ∈ Q(tf ), and thus

give rise to a semibounded self-adjoint operator Tf = U †S(∞)f U on the closure

Q(tf ) ⊂ L2a(gλ).

By the monotone convergence of forms the self-adjoint operators associatedwith s

(ν)f converge in the strong resolvent sense [37], which in turn implies strong

convergence of the semigroups they generate [34, Theorem S.14]. ✷THEOREM 6.4. Provided f and g are such that |a|2 ∈ Kloc and υ, f ∈ K±,and the conclusion of Proposition 5.3 holds, then for t > 0 the continuous integralkernel of the semigroup generated by S(ν)f converges in the limit ν → ∞ pointwise


to the kernel of the semigroup associated with the generator S(∞)f on E(L2(D)),

limν→∞ e−tS(ν)f (z, z′) = (e−tS(∞)f E)(z, z′). (45)

Proof. The proof heavily borrows from the strategy of [9] which is accommo-dated here to the case of unbounded f . The key to the present generalization is theuse of monotone form convergence. Fundamental to the proof is the well-knownFeynman–Kac–Itô formula [11] which expresses the continuous integral kernel of

the semigroup e−tS(ν)f as the path integral

e−tS(ν)f (z, z′) =∫

{TD>t}exp

{1

2

∫ t

0[a(b(s)) db(s) − a(b(s)) db(s)]−

−∫ t

0ds(νυ + f )(b(s))

}dµ(ν)z,0;z′,t , (46)

where in this case it does not matter whether the stochastic integral in the exponentis interpreted in the Itô sense or according to Fisk and Stratonovich [35], because∂1a1 + ∂2a2 = 0.

For notational convenience we fix a reference diffusion constant ν0 > 0 andabbreviate for α � 0, w ∈ C

η(α)w (z) := e−tS(ν0)αf (z,w). (47)

As preparation for the main part of the proof we state three properties of η(α)w :

(1) Each η(α)w is a bounded and continuous function [11, Theorem 4.1] and lies inL2(D), which follows from the inequality

‖η(α)w ‖ � supw∈D

‖η(α)w ‖ = ‖e−tS(ν0)αf ‖2,∞, (48)

where the last term is the finite operator norm of e−tS(ν)αf considered as mappingfrom L2(D) to L∞(D) [11, Estimate (2.39)].

(2) The mapping w �→ η(α)w is a strongly continuous mapping, because of theidentity (η(α)w , η

(α)

w′ ) = exp(−2tS(ν0)αf )(w,w

′) and due to the continuity of thekernel [11, Theorem 6.1].

(3) In addition, the mapping α �→ η(α)w is also strongly continuous. To see this, weemploy (46) to represent the difference of the two integral kernels in∫ |η(α)w (z) − η(α′)

w (z)|2 dλ(z) as one path integral. Now we can bound the ab-solute value of the path integrand by 2 exp(

∫ t0 (−ν0u + α0f

−) ds) with somelarge α0, which shows that dominated convergence applies in the limit α′ → α.

As the main part of the proof we show that

limν→∞ e−tS(ν)f (z, z′) = (κz, e−tS(∞)f Eκz′) (49)


which by an analogue of Corollary 3.6 in connection with Subsection 5.1 consti-tutes the continuous integral kernel for exp(−tS(∞)f )E on L2(D).

To see (49), we use the semigroup property and rewrite the integral kernel asscalar product

e−tS(ν)f (z, z′) = (η(ν0/ν)z , exp(−tS(ν−2ν0)

(ν−2ν0)f/ν)η(ν0/ν)

z′ ) (50)

which converges in the limit ν → ∞ to

limν→∞ e−tS(ν)f (z, z′) = (η(0)z , e−tS(∞)f E η

(0)z′ ). (51)

This can be deduced from the strong continuity of η(α)z in α and the strong con-vergence stated in Theorem 6.3 together with the uniform boundedness of theoperators exp(−tS(ν−2ν0)

(ν−2ν0)f/ν) according to the Banach–Steinhaus theorem.

To finish the proof, we observe that the right-hand side of (51) is an integralkernel for exp(−tS(∞)f )E on L2(D) which is, in addition, continuous in z and z′and therefore coincides with the right-hand side of (49). The continuity of (51) isclear, and with E exp(−tS(ν0)

0 ) = E it can be checked that it indeed constitutes anintegral kernel. ✷COROLLARY 6.5. Combined with the Feynman–Kac–Itô formula (46) and theidentity

e−tTf (z, z′) = (U †e−tS(∞)f EU)(z, z′), (52)

the result (45) provides a probabilistic expression for the continuous integral kernelof the semigroup generated by Tf on Q(tf ) ⊂ L2

a(gλ). This is the generalizedDaubechies–Klauder formula, which states that for admissible f and g and fort > 0,

e−tTf (z, z′)

= limν→∞

√(z′)g(z)

∫{TD>t}

exp

{1

2

∫ t

0[a(b(s)) db(s) − a(b(s)) db(s)]−

−∫ t

0ds(νυ + f )(b(s))

}dµ(ν)z,0;z′,t . (53)

In particular, the choice f = 0 yields the reproducing kernel of the weightedBergman space which g characterizes.

Remarks 6.6. According to [8, 9], due to the significance of ν the expressionfor the integral kernel of the Berezin–Toeplitz semigroup in (53) is called an ultra-diffusive limit. By rescaling the Brownian bridge as in [9, Equation (11)], one


Table I.

No. D g(z) a(z) υ(z)

1 C1π e−|z|2 iz −2

2 C2j+1π (1 + |z|2)−2j−2 2(j+1)iz

1+|z|2 − 4(j+1)(1+|z|2)2

3(a) {|z|2 < 1} 2k−1π (1 − |z|2)2k−2 2(k−1)iz

1−|z|2 − 4(k−1)(1−|z|2)2

3(b) C2|z|2k−1

π+(2k) K2k−1(2|z|) Z2k−2izZ2k−1

− 2Z2

2k−1(Z2k−2Z2k−1 +|z|2(Z2

2k−2 −Z2k−1Z2k−3))

may alternatively restate (53) as a long-time limit. Another version of the resultpointed out there implies that the integrand in (53) can be re-expressed in terms ofa complex Itô stochastic integral. All these versions will be omitted here.

Reading from the right to the left, (53) can be interpreted as a quantizationformula, which constructs from the functions f and g the semigroup generatedby Tf and thereby specifically selects the relevant Hilbert space Q(tf ) on whichTf is properly defined as a self-adjoint operator. In the context of quantization, ln gis interpreted as a Kähler potential on D. To our knowledge, without additionalsymmetry requirements, the setting considered here still lacks the proof of a cor-respondence principle [10, 17], for which probabilistic techniques might providehelpful tools.

EXAMPLES 6.7. In Table I we revisit the examples from Section 3 again and listthe functions a and υ belonging to each weight function g considered there. In thelast row we used the abbreviation Zσ := |z|σKσ (2|z|).

The following remarks examine the validity of Equation (53) for each case.

Remarks 6.8. The primary condition for the validity of (53) in a specific situa-tion is that the weight function leads to admissible a and υ. If this is satisfied, anyKato-decomposable symbol f allows a probabilistic expression for the semigroupgenerated by Tf .

By inspection we decide whether each example can be used in the formula (53).For the examples 1 and 2 the function a is continuous and υ is even bounded, hencethey admit the probabilistic representation according to (53).

Unfortunately, for the Example 3(a) either k = 1 and Proposition 5.3 is notsatisfied or the scalar potential υ is not Kato decomposable because of a strongnegative singularity towards the boundary of the disc. Therefore the probabilisticrepresentation is not valid in either case. However, the desired monotone form


convergence can be recovered with the modification of the domain suggested inRemark 5.2 which is associated with Neumann boundary conditions.

On the other hand, the vector and scalar potentials a and υ emerging from theBarut–Girardello representation 3(b) are admissible for k > 1, which can be readoff from the asymptotics:

limr→0

r |σ |Kσ(2r) = 1

2

∫ ∞

0t−σ−1e−1/t dt

for σ �= 0, and limr→0K0(2r)/ ln 2r = −1.Within the setting of Example 1 the identity (53) is in essence a result by [14].

The identity corresponding to Example 2 has already been worked out in [9], whereit is also compared to a similar formula derived in [14]. These last two alternativesare among the few known ways to obtain a mathematically well-founded path in-tegral for spin. The weighted Bergman space of Example 3(b) was not previouslyknown to admit a formula of type (53), which can now serve as an alternative topath-integral formulas that do not contain genuine path measures [18, 22].

7. Conclusion

In this paper we have tried to indicate the key principles behind the construction ofDaubechies and Klauder and have thus derived a natural generalization. As a majorspin-off we developed criteria for self-adjointness of Berezin–Toeplitz operators.Once the relation to Schrödinger operators is established, this is an immediateconsequence.

As to further ramifications, we point out that with a suitable analyticity argu-ment one could obtain from (53) the probabilistic expression for the unitary groupe−itTf which was a primary motivation for [14, 15]. It might also be interestingto investigate random Berezin–Toeplitz operators, for which the probabilistic rep-resentation seems to offer an appropriate analytic framework. Finally, it deservesmentioning that the concept of path transformations is also applicable in the contextof (53) in order to relate the resolvents of certain Berezin–Toeplitz operators (inpreparation).

Acknowledgements

It is a pleasure to thank Hajo Leschke for his scientific guidance through the initialstage and Simone Warzel for her valuable criticism and participation in the strugglefor the clear picture. Thanks are extended to John R. Klauder for encouragement,inspiration and lots of resourceful advice. I am also indebted to Kazuyuki Fujii fordrawing my attention to the Barut–Girardello representation. The Studienstiftungdes deutschen Volkes is acknowledged for financial support.


References

1. Alicki, R. and Klauder, J. R.: Quantization of systems with a general phase space equippedwith a Riemannian metric, J. Phys. A 29 (1996) 2475–2483.

2. Alicki, R., Klauder, J. R. and Lewandowski, J.: Landau-level ground state and its relevance fora general quantization procedure, Phys. Rev. A 48 (1993), 2538–2548.

3. Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform,Part I, Comm. Pure Appl. Math. 14 (1961), 187–214.

4. Barut, A. O. and Girardello, L.: New ‘coherent’ states associated with non-compact groups,Comm. Math. Phys. 21 (1971), 41–55.

5. Bar-Moshe, D. and Marinov, M. S.: Berezin quantization and unitary representation of Liegroups, In: R. L. Dobrushin, R. L. Minlos, M. A. Shubin and A. M. Vershik (eds), Topics in Sta-tistical and Theoretical Physics, Amer. Math. Soc. Transl. 177, Amer. Math. Soc., Providence,1996, pp. 1–21.

6. Berezin, F. A.: Quantization, Math. USSR-Izvest. 8 (1974), 1109–1165. Russ. orig.: Izv. Akad.Nauk SSSR Ser. Mat. 38 (1974), 1116–1175.

7. Bergman, S.: The Kernel Function and Conformal Mapping, 2nd edn, Amer. Math. Soc.Survey 5, Amer. Math. Soc., Providence, 1970.

8. Bodmann, B., Leschke, H. and Warzel, S.: A rigorous path-integral formula for quantumspin via planar Brownian motion, In: R. Casalbuoni, R. Giachetti, V. Tognetti, R. Vaiaand P. Verrucchi (eds), Path Integrals from peV to TeV, World Scientific, Singapore, 1999,pp. 173–176.

9. Bodmann, B., Leschke, H. and Warzel, S.: A rigorous path integral for quantum spin usingflat-space Wiener regularization, J. Math. Phys. 40 (1999), 2549–2559.

10. Bordemann, M., Meinrenken, E. and Schlichenmaier, M.: Toeplitz quantization of Kählermanifolds and gl(n), n → ∞ limits, Comm. Math. Phys. 165 (1994), 281–296.

11. Broderix, K., Hundertmark, D. and Leschke, H.: Continuity properties of Schrödinger semi-groups with magnetic fields, Rev. Math. Phys. 12 (2000), 181–225.

12. Cichon, D.: Notes on unbounded Toeplitz operators in Segal-Bargmann spaces, Ann. Polon.Math. 64 (1996), 227–235.

13. Cycon, H. L., Froese, R. G., Kirsch, W. and Simon, B.: Schrödinger Operators, withApplication to Quantum Mechanics and Global Geometry, Springer, Berlin, 1987.

14. Daubechies, I. and Klauder, J. R.: Quantum-mechanical path integrals with Wiener measure forall polynomial Hamiltonians II, J. Math. Phys. 26 (1985), 2239–2256.

15. Daubechies, I. and Klauder, J. R.: True measures for real time path integrals, In: M. L. Gutz-willer, A. Inomata, J. R. Klauder and L. Streit (eds), Path Integrals from meV to MeV, BielefeldEncounters in Phys. Math., World Scientific, Singapore, 1986, pp. 425–432.

16. Daubechies, I., Klauder, J. R. and Paul, T.: Wiener measures for path integrals with affinekinematic variables, J. Math. Phys. 28 (1987), 85–102.

17. Engliš, M.: Asymptotics of the Berezin transform and quantization on planar domains, DukeMath. J. 79 (1995), 57–76.

18. Fujii, K. and Funahashi, K.: Extension of the Barut–Girardello coherent state and path integral,J. Math. Phys. 38 (1997), 4422–4434.

19. Gradshteyn, I. S. and Ryzhik, I. M.: Tables of Integrals, Series, and Products, 5th edn,Academic Press, Boston, 1995.

20. Hinz, A. M.: Regularity of solutions for singular Schrödinger equations, Rev. Math. Phys. 4(1992), 95–161.

21. Hinz, A. M. and Stolz, G.: Polynomial boundedness of eigensolutions and the spectrum ofSchrödinger operators, Math. Ann. 294 (1992), 195–211.

22. Inomata, A., Kuratsuji, H. and Gerry, C. C.: Path Integrals and Coherent States of SU(2) andSU(1, 1), World Scientific, Singapore, 1992.


23. Janas, J. and Stochel, J.: Unbounded Toeplitz operators in the Segal–Bargmann space, II,J. Funct. Anal. 126 (1994), 419–447.

24. Kato, T.: Schrödinger operators with singular potentials, Israel J. Math. 13 (1973), 135–148.25. Klauder, J. R.: Quantization is geometry, after all, Ann. Phys. 188 (1988), 120–141.26. Klauder, J. R.: Quantization on non-homogeneous manifolds, Internat. J. Theor. Phys. 33

(1994), 509–522.27. Klauder, J. R. and Onofri, E.: Landau levels and geometric quantization, Internat. J. Modern.

Phys. 4 (1989), 3939–3949.28. Leinfelder, H. and Simader, C. G.: Schrödinger operators with singular magnetic vector

potentials, Math. Z. 176 (1981), 1–19.29. Maraner, P.: Landau ground state on Riemannian surfaces, Modern. Phys. Lett. A 7 (1992),

2555–2558.30. Meschkowski, H.: Hilbertsche Räume mit Kernfunktion, Springer, Berlin, 1962.31. Neeb, K. H.: Coherent states, holomorphic extensions, and highest weight representations,

Pacific J. Math. 174 (1996), 497–541.32. Perelomov, A.: Generalized Coherent States and their Application, Springer, Berlin, 1986.33. Reed, M. and Simon, B.: Methods of Modern Mathematical Physics, vol. II, Fourier Analysis,

Self-Adjointness, Academic Press, New York, 1975.34. Reed, M. and Simon, B.: Methods of Modern Mathematical Physics, vol. I, Functional Analysis,

Academic Press, New York, 1980.35. Rogers, L. C. G. and Williams, D.: Diffusions, Markov Processes, and Martingales, vol. 2: Itô

calculus, Wiley, Chichester, 1987.36. Revuz, D. and Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn, Springer,

Berlin, 1999.37. Simon, B.: A canonical decomposition for quadratic forms with applications to monotone

convergence, J. Funct. Anal. 28 (1978), 377–385.38. Simon, B.: Functional Integration and Quantum Physics, Academic Press, New York, 1979.39. Simon, B.: Maximal and minimal Schrödinger forms, J. Oper. Theory 1 (1979), 37–47.40. Simon, B.: Schrödinger semigroups, Bull. Amer. Math. Soc. (NS) 7 (1982), 447–526; Erratum:

ibid. 11 (1984), 426.41. Weidmann, J.: Linear Operators in Hilbert Spaces, Springer, New York, 1980.


307

Geometrical Lagrangian for a SupersymmetricYang–Mills Theory on the Group Manifold

M. F. BORGESDepartment of Mathematics and Applied Mathematics, University of Cape Town, Private Bag 7701Rondebosch, Cape Town, South Africa and UNESP, State University of São Paulo, Departmentof Computing, 15054-000, São José do Rio Preto, Brazil. e-mail: [email protected]

(Received: 12 April 2001; in final form: 14 November 2001)

Abstract. Perhaps one of the main features of Einstein’s General Theory of Relativity is that space-time is not flat itself but curved. Nowadays, however, many of the unifying theories like superstringson even alternative gravity theories such as teleparalell geometric theories assume flat spacetime fortheir calculations. This article, an extended account of an earlier author’s contribution, it is assumed acurved group manifold as a geometrical background from which a Lagrangian for a supersymmetricN = 2, d = 5 Yang–Mills – SYM, N = 2, d = 5 – is built up. The spacetime is a hypersurfaceembedded in this geometrical scenario, and the geometrical action here obtained can be readilycoupled to the five-dimensional supergravity action. The essential idea that underlies this work has itsroots in the Einstein–Cartan formulation of gravity and in the ‘group manifold approach to gravityand supergravity theories’. The group SYM, N = 2, d = 5, turns out to be the direct product ofsupergravity and a general gauge group G: G = G ⊗ SU(2, 2/1).

Mathematics Subject Classifications (2000): 83E50, 81T13.

Key words: group manifold, supergravity, supersymmetry, super Yang–Mills theory.

1. Introduction

The understanding of low energy phenomena may indicate the need for field the-ories with gauge symmetries as a convenient scheme to describe the fundamentalinteractions between elementary particles. In many ways, physics is dominated bythe striking successes of quantum electrodynamics and the trends in the descriptionof fundamental interactions (chromodynamics and Glashow–Weinberg–Salam the-ory). As a result of this, the theory of gravitation is sometimes required to conformto the principles and fashions prevalent in elementary particle physics. Therefore,it seems logical that all gauge theories could be interpreted in only one fundamen-tal principle. In other words, the different symmetries related to the fundamentalinteractions at a low level of energy could be the fragments of a higher symmetry,related to interactions at a more elevated level of energy. Supergravity arises natu-rally in such a picture. Supergravity theories are an embedding of Einstein’s theoryof gravity in a broader framework, with two main aims: renormalization (or bet-ter, finiteness of quantum gravity) and (super) unification with other interactions.

308 M. F. BORGES

In such theories, the internal symmetries (gauge groups of Yang–Mills) and thespacetime symmetries (gauge groups of Poincaré) are unified into one algebraicstructure. In this way also the internal symmetries show a geometrical feature thatextend the geometrical interpretation of pure gravity. A gauge structure for thetheory of gravitation was carried out by Cartan [1] in the twenties and followed byKibble and Sciama in the sixties in an attempt of treat Einstein’s gravitation as aYang–Mills theory.

The so-called Cartan–Kibble–Sciama gauge formulation of gravity was extend-ed by Regge and Ne’eman [3] and soon after by Regge and collaborators [4], inTurin, in an approach called ‘group manifold approach to supersymmetric theo-ries’.

In this article, we will give an extended account of that approach and of previousresults obtained by the author in [5] and by Borges and Masalskiene in [6], in theattempt to construct an extended gravity theory of the Yang–Mills type. With thisaim, a geometrical Lagrangian for the N = 2, d = 5 supersymmetric Yang–Millstheory, where d stands for the dimension of the spacetime manifold and N is thenumber of generations of supersymmetry transformations, is built up on the group-manifoldG. The procedure to obtain the Lagrangian in this paper is, in many ways,an application of the general scheme provided by Castellani, D’Auria and Fré [7],and by the fact that the theory is directly coupled to the N = 2, d = 5 supergravityon the group manifold [8], that means G = G ⊗ SU(2, 2/1), where G is a generalgauge group.

In general, the construction of a Lagrangian for supersymmetric theories is acomplicated task. Although a geometrical first-order N = 2 supergravity action infive dimensions was established some time ago [8, 9], a minimal off-shell second-order supergravity version has only recently been developed [10]. There are wellknown formulations for d = 4, N = 2 [11, 12] as well as for d = 6, N = 2 [13],and even a geometrical and generic first-order N = 2 Yang–Mills case obtainedby Fré et al. [14]. However, a specific N = 2, d = 5 first-order Yang–Mills theoryreadily coupled to supergravity needs to be worked out. We intend to bridge thatgap by constructing such a geometric coupling Yang–Mills action.

2. Geometrical Action for a Supersymmetric Five-DimensionalYang–Mills Theory

The geometrical supersymmetric Yang–Mills theory is said to be ‘impure’, conta-minated by the presence of 0-form matter fields. In a ‘pure’ theory, instead, theonly fields present are the 1-forms µA, which constitute the so-called pseudo-connection. Geometrical Yang–Mills theories have another feature: they can benaturally coupled to supergravity, if the latter is a pure theory. For instance, theglobally supersymmetric N = 2, d = 5 Yang–Mills becomes locally supersym-metric in the geometrical formalism by adjoining the action of the N = 2, d = 5supergravity which is a ‘pure’ theory of supergravity. Two motivations evidenc-

A GEOMETRICAL LAGRANGIAN FOR A SYM THEORY 309

ing the importance of supersymmetric Yang–Mills theories within the context ofunification, may be cited as follows: (i) their possible connection with other cases(for example, with pure supergravity theories formulated in higher-dimensionalspacetimes) through the technique of dimensional reduction, thereby improvingthe knowledge of the various supersymmetric actions; (ii) the second motivationleads us to quantization: it is well known that when the couplings – matter cou-pled to supergravity – are just those which arise naturally when a higher extendedsupergravity theory is considered as a theory with less symmetry, then part of whatwas the gravitational multiplet appears in a particular way which reflects the largersymmetry of the theory.

A general hypothesis for the curvature of the N = 2, d = 5 Yang–Mills theoryhas then proposed in [5] as the following:

dA = FabV a ∧ V b + iλA ∧ �mψA ∧ Vm + if σ ∧�A ∧�A;DλA = �mAV

m − iFab ∧�ab�A − φa ∧ �a�A,

Dσ = φa ∧ V a + i

2λA ∧�A;

DF ab = Gabm ∧ V m + i�A

[a ∧ �b]�A, (1)

where V is the Vierbein associated to the graviton, � is the graviton field, λ is theDirac spinorial field, and σ the scalar field. F stands for the curvature associatedwith the gauge group G.

Determination through the analysis of the Bianchi identities of a compatiblesystem of equations for the parameters of the curvature, indicated that the hypoth-esis made for them is acceptable. The group manifold G of this theory is thendetermined by the fact that the theory is directly coupled to supergravity, whichmeans

G = G ⊗ SU(2, 2/1). (2)

The theory has also presented a bundle structure where H ′ = G ⊗ SO(1, 4) ⊗U(1) would be the fibre and the quotient space, G/H ′, the base space of theprincipal fibre, identified with the superspace.

Besides its symmetry group properties, this theory has presented another im-portant feature. The Bianchi identities are satisfied as both the Dirac equation andthe homogeneous Maxwell equations hold [5]:

�m�mA = 0; G[ab/m] = 0. (3)

This result is related to the closure of the algebra of supersymmetry transfor-mations ‘on shell’. Finally, through the explicit determination of the curvature bymeans of the Bianchi identities, we are able to build up from now onwards thegeometrical action of this N = 2, d = 5 supersymmetric Yang–Mills theory,which will become readily coupled to the N − 2, d = 5 supergravity. Beforegetting on with that, however, some basic requirements must be taken into accountas preliminary ‘precautions’:

310 M. F. BORGES

(1) the action is an integral of 5-forms developed on an arbitrary hypersurface M5

(5 = number of dimensions of spacetime), immersed in the whole manifold G,(2) the action must be stationary relating to the variations of fields, and to the varia-

tions of the hypersurface M5. It implies that the equations of motion interpretedin differential forms and resulting from the variations of the Lagrangian relatedto the fields of the theory will have validity on the whole manifold,

(3) the 5-form Lagrangian is gauge invariant under the group

G⊗ SO(1, 4)⊗ U(1) (4)

with a general form given by

LYM = �A + νAθA + νABθ

A ∧ θB, (5)

where �A, νA, νAB are polynomials with constant coefficients and θA, θB are the‘curvatures’ of the manifold for the ‘purely’ Yang–Mills case.

The Lagrangian, LYM will be built up with all the fields present in the theoryin first-order form: A, λA, σ, Fab, φ

a, Va,�A,�A. The Hodge operator will not beused. We will assume, for simplicity, throughout the calculations that the group Gis Abelian. Such a requirement, however, doesn’t constitute any over-simplificationfor the theory. In fact, we have that

DF = DdA+D[A,A] = DdA, as D[A,A] = D(( 12CθνA

θ ∧ Aν)) = 0.

Consequently, the terms in [A,A] do not make any contribution to the parametriza-tion of the curvatures.

(4) LYM is not trivial. That means that if curvatures of supergravity are zero, thenthe equations of motion must be satisfactorily worked out, having as con-tent both the curvature of the gauge group G and of the covariant derivativesDλA,Dσ ,DF

ab, whose parametrization is already completed (1).(5) Finally, we would require that the projection of the equations of motion on

spacetime will lead us to the equations of Maxwell, Dirac and Klein–Gordon(usually present in a simple version of a Yang–Mills theory in four dimen-sions).

The N = 2, d = 5 Yang–Mills action, AYM , can then be written as

AYM =∫M5

LMaxwell + LDirac + LKlein–Gordon ++ others terms to be determined, (6)

where LMaxwell, LDirac, LKlein–Gordon are the Lagrangians of Maxwell, Dirac andKlein–Gordon written in first-order formalism and in five dimensions. The Maxwellaction is, in second-order formalism, written as

AMaxwell =∫FuvF

uv√−gd5X. (7)


The last, as presented, is useless for our purposes. The fields Fab and A should betreated independently as, for example, in the first-order formalism. In the secondorder one, Fab is related to A by

Fab = 1

2(∂aAb − ∂bAa). (8)

Consequently, the problem is reduced to finding out through the Maxwell action,in five dimensions and in first-order formalism, the relation between A and Fab, butmaking no use of the Hodge operator. That will lead us to a new term to be addedto (8) in its first-order version.

The so-called first-order version of the Maxwell action (8) is written as

AMaxwell =∫FabF

abV i ∧ V j ∧ V K ∧ V l ∧ V mεijklm, (9)

where

V i = V iµ dxµ; (10)

gµν = V µa V

νb η

ab; (11)

detV = √−g; (12)

εµνρσ�d5x = dxµ ∧ dxν ∧ dxρ ∧ dxσ ∧ dx�; (13)

d5x√−g = V i ∧ V j ∧ V k ∧ V l ∧ Vmεijklm (14)

and ∧ is the edge product.The new, hypothetical Maxwell action is then taken to be (with the new term

mentioned above, added in):

AMaxwell =∫

dAF abV c ∧ V d ∧ V eεabcde ++KF abF

abV i ∧ V j ∧ V k ∧ V l ∧ V mεijklm. (15)

The variation of (15) to F ab produces

dA ∧ V c ∧ V d ∧ V eεabcde ++2KF abV

i ∧ V j ∧ V k ∧ V l ∧ V mεijklm = 0; (16)

dA = XlmVl ∧ Vm + (terms with �A). (17)

In projecting (16) on the five-dimensional spacetime (= terms in V∧V∧V∧V∧V ),we obtain that

K = (− 120

); Xlm = Flm. (18)

There, in spacetime M5, we find out as desired the correct relation between Fab andA, given by

dA = FabVa ∧ V b = Fµν dxµ dxν = DµAν dxµ dxν. (19)

312 M. F. BORGES

Consequently, our Yang–Mills action (partial) assumes the following form

AYM =∫

dAF abV c ∧ V d ∧ V eεabcde ++ (− 1

20

)FabF

abV i ∧ V j ∧ V k ∧ V mεijklm ++ other terms to be determined. (20)

The Dirac term in the above Lagrangian, in first-order formalism, may be writ-ten as

iλA�aDλA ∧ V b ∧ V c ∧ V d ∧ V eεabcde. (21)

In fact the projection of it on the spacetime gives us the usual Dirac Lagrangian:

iλA�mDmλA, (22)

where

DλA = DmλAVm + (terms with �A). (23)

The following terms to be found are those related to the Klein–Gordon La-grangian. The procedure taken is analogous to that for the Maxwell case. In thefirst-order formalism, the fields σ and φa are considered as independent fields. Theconnection between them should arise out of the equation of motion.

In the action (20), the following terms corresponding to the Klein–Gordonaction will be introduced;

AKlein–Gordon =∫ (

12

)φaφaεijklmV

i ∧ V j ∧ V k ∧ V l ∧ V m ++WφadσV b ∧ V c ∧ V d ∧ V eεabcde. (24)

The variation of (24) related to φa in considering

Dσ = ZmVm + (terms with �A), (25)

is such that the projection of this result on the spacetime will give us

W = −5 (26)

and

Za = φa. (27)

Consequently,

dσ = φaVa = DaσV

a. (28)


The action (24) is then implying that φa is the covariant derivative of σ . Withthe last steps taken into account, the Lagrangian of Yang–Mills (20) may then bewritten as

AYM =∫

dAF ab ∧ V e ∧ V d ∧ V eεabcde −− (

120

)FabF

abV i ∧ V j ∧ V k ∧ V l ∧ Vmεijklm ++ (

12

)aφaφ

aV i ∧ V j ∧ V k ∧ V l ∧ V mεijklm −− 5aφadσV b ∧ V c ∧ V d ∧ V eεabcde ++ ibλA�

aDλA ∧ V b ∧ V c ∧ V d ∧ V eεabcde ++ other terms to be added. (29)

a and b are parameters to be determined.To find out all the other remaining terms of the Yang–Mills action, the starting-

point will be the partial action (29), with the terms of Maxwell, Dirac, and Klein–Gordon already identified.

Let us consider (29). The variation of it relating to Fab, leads, after substitutionof dA as indicated in (1), to the following equation:

FlmVl ∧ V m ∧ V c ∧ V d ∧ V eεabcde++iλA�m�A ∧ V m ∧ V c ∧ V d ∧ V eεabcde++iσ�A ∧�A ∧ V c ∧ V d ∧ V eεabcde+− (

110

)FabV

i ∧ V j ∧ V k ∧ V lεijklm = 0. (30)

Consequently, to avoid the possibility that the only solution for λA and σ is atrivial one, we must add new terms to the action. These will be constructed onlyfrom the fields of the theory and consistent, after their variations related to Fab, φaand λA, in such a way as to make (30) identically zero, but without having zero asa solution.

Hypothetically, we will introduce in (29) terms of the following type:

−iF abλA�m�A ∧ Vm ∧ Vc ∧ Vd ∧ Veεabcde; (31)

−iF abσ�A ∧�A ∧ V c ∧ V d ∧ V eεabcde; (32)(52

)iaφaλA�A ∧ V b ∧ V c ∧ V d ∧ V eεabcde; (33)

cdAλA�ab�AVa ∧ V b; (34)

edσλA�ab�AVe ∧ Vd ∧ Veεabcde; (35)

if λA�abλAλA ∧�B ∧ V c ∧ V d ∧ V eεabcde; (36)

igλAλBδAC�c�ab�B ∧ V c ∧ V d ∧ V eεabcde; (37)

ihλA�abλA�B�

c�B ∧ Va ∧ Vb ∧ Vc; (38)

ilλA�aλBδACX

bcBC ∧ Va ∧ Vb ∧ Vc, (39)

where

XbcBC = �C�

bc�B.

314 M. F. BORGES

The final expression resulting from the variation of (29) related to λA with allthe terms mentioned above, will be the following:

iF ab�m�AVm ∧ V e ∧ V d ∧ V eεabcde+

+(52

)iaφa�AV

b ∧ V c ∧ V d ∧ V eεabcde++2ib�a�mAV

m ∧ V b ∧ V c ∧ V d ∧ V eεabcde++24bFab�cd�AV

a ∧ V b ∧ V c ∧ V dεabcde++2ibF am�m�AV

b ∧ V c ∧ V d ∧ V eεabcde−−2ibφa�A ∧ V b ∧ V c ∧ V d ∧ V eεabcde−−4bφm�

am�AVb ∧ V c ∧ V d ∧ V eεabcde−

−2b�aλA�B�b�B ∧ V c ∧ V d ∧ V eεabcde+

+cFab�cd�AVa ∧ V b ∧ V c ∧ V d+

+icσ�ab�A�B ∧�B ∧ V a ∧ V b−−(

18

)ciλBδAC�C�

cd�B ∧ Vm ∧ V r ∧ V sεcdmrs++(

14

)ci�mλBδAC�C�

ab�B ∧ V m ∧ Va ∧ Vb−−(

12

)c�bsλBδAC�C�

cd�B ∧ V m ∧ V � ∧ Vbεmcdrs−−(

14

)c�rsλBδAC�C�

cd�B ∧ Vc ∧ V a ∧ V bεdabrs++(

18

)ci�bcλA�B�a�B ∧ V c ∧ V a ∧ V b+

+(1

16

)c�rλA�B�

m�B ∧ V s ∧ V a ∧ V bεmabrs−−(

116

)c�rsλA�B ∧�B ∧ V m ∧ V a ∧ V bεmabrs−

−eφl�ab�AVlV e ∧ V d ∧ V eεabcde+

+(18

)eiλBδAC�C�

ab�B ∧ V c ∧ V d ∧ V eεabcde++(

34

)ie�KλBδAC�C�

ab�B ∧ Va ∧ Vb ∧ Vk++(

12

)e�bmδAC�C�

am�BVc ∧ V d ∧ V eεabcde+

+(38

)ei�lmλA�B�

c�BVc ∧ Vl ∧ V m−−(

116

)e�bλA�B�c�BVa ∧ Vd ∧ Veεabcde+

+2if�abλA�B�BVe ∧ V d ∧ V eεabcde+

+2igλBδAC�C�ab�BV

c ∧ V d ∧ V eεabcde++2ih�abλA�B�

c�BVa ∧ Vb ∧ Vc++2il�cλB�BδAC�C�

ab�BVc ∧ Va ∧ Vb++(

116

)ei�abλA�B�BVc ∧ Vd ∧ Veεabcde = 0. (40)

From a final analysis of Equation (40), we find the presence of only one term withthree �,

−icσ�ab�A�B ∧�B ∧ V a ∧ V b. (41)


That term will imply further inclusion in the action (40) of another term, again toavoid a trivial solution. This term will be

icσλA�ab�A�B ∧�B ∧ V a ∧ V b. (42)

Finally, by projecting terms of the same type as in (44), for convenient tangentvectors, one obtains the following set of equations for the parameters:

1 − 4b = 0, 3e − c = 0,(

52

)a − 2b = 0, c + 24b = 0,

e + 8b = 0, −2b − (1

16

)c − (

116

)e = 0,

(18

)c + (

38

)e + 2h = 0,

−(116

)c + (

116

)e + 2f = 0, −(

18

)c + (

18

)e + 2g = 0,(

14

) − c + (34

) + 2l = 0. (43)

From these equations unique values for the parameters could be obtained:

b = 14; e = −2; g = − 1

4;a = 1

5 ; h = 34 ;

c = −6; f = − 18 ; l = 3

2 . (44)

Further variations of (35) related to σ and A, with all the new terms previouslydiscussed, would lead us to the introduction of other terms in the action, to avoidtriviality. Such terms are

imdA ∧ σ ∧�A�m�A ∧ Vm, (45)

and

pσ 2�A�m�A�B ∧�B ∧ Vm, (46)

from the variation related to σ , and

inAdA ∧�A ∧�A, (47)

from the variation related to A.The parameters m, p and n are determined without any incompatibility. Their

values are

m = −6, n = 3

2and p = −3. (48)

The Yang–Mills action, taking into account all the terms reported here and thecalculated parameters, will be

AYM =∫M5(dAF abV c ∧ V d ∧ V eεabcde −

− (120

)FabF

abV i ∧ V j ∧ V k ∧ V l ∧ Vmεijklm+

316 M. F. BORGES

+ (1

10

)φaφaV

iV j ∧ V k ∧ V l ∧ V mεijklm −−φadσV b ∧ V c ∧ V d ∧ V eεabcde ++ (

14

)λA�

aDλAVb ∧ V c ∧ V d ∧ V eεabcde −

− iF abλA�m�AVm ∧ V c ∧ V d ∧ V eεabcde −

− iF abσ�A�AVc ∧ V d ∧ V eεabcde +

+ (12

)iφaλA�AV

b ∧ V c ∧ V d ∧ V eεabcde −− 6dAλA�ab�AV

a ∧ V b −− 2(dσ )λA�

ab�AVc ∧ V d ∧ V eεabcde −

− 6iσλA�ab�A�B�BVa ∧ V b −

− (18

)iλA�

abλA�B ∧�B ∧ V c ∧ V d ∧ V eεabcde −− (

14

)iλAλBδAC�C�

ab�B ∧ V c ∧ V d ∧ V eεabcde ++ (

34

)iλA�

abλA�B�c�B ∧ Va ∧ Vb ∧ Vc +

+ (32

)iλA�

cλBδAC�c�ab�B ∧ Vc ∧ Va ∧ Vb −

− 6idAσ(�A�m�A) ∧ Vm − 3σ 2�A�

m�A ∧�B ∧�B ∧ Vm ++ (

32

)iA ∧ dA ∧�A ∧�A). (49)

3. Conclusions

In this paper, the author has assumed a curved group-manifold as the geometricalscenario from which a classical alternative gravity theory of the Yang–Mills typeis considered in a fuller account, completing some results obtained in an earlierpaper [5]. The spacetime is presented as a hypersurface embedded in this geo-metrical background. This work is deeply routed in a previous article by Reggeand Ne’eman [3]. Nevertheless, here dynamics is controlled by geometry in thesense that first the curvatures and Bianchi identities were established [5, 6], andthen the Lagrangian and motion field equations worked out. Using the basic toolsof differential geometry, exterior forms and exterior derivative, gravitation andits extension as a Yang–Mills theory is then described over a group manifold G,G = G⊗SU(2, 2/1). The group G has the same relationship to the Poincaré groupas curved spacetime does to Minkovski spacetime, except that the existence of ametric is now replaced by the use of the vierbein.

So far, one of the main motivations standing behind the present work is theclose relationship that today connects quantum gravity and Yang–Mills theoriesin the nonperturbative strings theory scenario, and 11-dimensional supergravityframework, as exemplified by many others such as Lukas et al. [15], Green [16],and in the Gunaydin and Zagermann [17] formulation of d = 5, N = 2 mattercoupled supergravity, where the physical theory on spacetime can also be recoveredby truncating the supergravity sector. The author expects to present a quantum


version of the main results here reported, which is still under investigation, in thenear future.

Acknowledgements

It is a great pleasure to thank George Ellis, Brian Hahn and Di Loureiro for theirhospitality in Cape Town, South Africa. I should also thank George Ellis for abrief comment on Sciama’s [2] earlier contribution towards describing all forces ofnature by means of non-Riemannian geometries, and the possible physical contentof those in terms of a Vierbien formalism.

References

1. Cartan, E.: Sur les varietes a connexions affines et la theorie de la relativité generalisée, Ann.Ecole Norm. Sup. 40 (1923), 325; 41 (1924), 1; 42 (1925), 3; Reprinted in Oeuvres completes,vol. 3, Gauthier-Villars, Paris, 1955. English version: On Manifolds with an Affine Connectionand the Theory of General Relativity, Bibliopolis, Bologna, Italy, 1986.

2. Kibble, T. W.: Lorentz invariance and the gravitational field, J. Math. Phys. 2 (1961), 212;Sciama, D. W.; Les bases physiques de la théorie du champ unifié, Ann. Inst. H. Poincaré 17(1961), 1.

3. Ne’eman, Y. and Regge, T.: Gauge theory of gravity and supergravity on a group manifold, Riv.Nuovo Cimento 1 (1978), 5.

4. D’Adda, A., D’Auria, R., Fré, P. and Regge, T.: Geometrical formulation of supergravitytheories on orthosympletie supergroup manifold, Riv. Nuovo Cimento 3 (1980), 6.

5. Borges, M. F.: Bianchi identities for an N = 2, d = 5 supersymmetric Yang–Mills theory onthe group manifold, J. Geom. Phys. 20 (1996), 142.

6. Borges, M. F. and Masalskiene, S. R. M.: Geometric extended gravity theory of Yang–Millstype, In: The Eighth Marcel Grossmann Meeting, Proc. meeting held at the Hebrew Univ. ofJerusalem, 22–27 June 1997, World Scientific, Singapore, 1999.

7. Castellani, L., D’Auria, R. and Fré, P.: Supergravity and Superstrings. A Geometric Perspective,vol. 1, Mathematical Foundations, World Scientific, Singapore, 1991.

8. D’Auria, R., Fré, P., Maina, E. and Regge, T.: Geometrical first order supergravity in 5 space-time dimensions, Ann. Phys. 135 (1981), 237.

9. D’Auria, R., Fré, P., Maina, E. and Regge, T.: A new group theoretical technique for the analysisof Bianchi identities and its application to the auxiliary field problem of d = 5 supergravity,Ann. Phys. 139 (1982), 93.

10. Zucker, M.: Minimal off-shell supergravity in five dimensions, Nuclear Phys. B 570 (2000),267.

11. Breitenlohner, P. and Sohnius, M. F.: Superfields, auxiliary fields and tensor calculus forN = 2extended supergravity, Nuclear Phys. B 165 (1980), 483.

12. De Wit, B., Van Holten, J. and Van Proeyen, S. A.: Transformation rules of N = 2 supergravitymultiplets, Nuclear Phys. B 167 (1980), 186.

13. Bergshoeff, E., Sezgin, E. and Salan, A.: Supersymmetric R2 actions, conformal invarianceand the Lorents Chern–Simons term in 6 and 10 dimensions, Nuclear Phys. B 279 (1987), 659.

14. Andrianopoli, L., Bertolini, M., Ceresole, A., D’Auria, R., Ferrara, S., Fré, P. and Magri, T.:N = 2 supergravity andN = 2 super-Yang–Mills theory on general scolar manifolds: simpleticcovariance, gaugings and the mommentum map, J. Geom. Phys. 23(2) (1997), 111.

15. Lukas, A., Ovrut, B. A., Stelle, K. S. and Waldran, D.: Heterotic M-theory in five dimensions,hep-th/9806051.

318 M. F. BORGES

16. Green, M. B.: Superstrings, M theory and quantum gravity, Classical Quantum Gravity,Millenium Issue, 16 (1999), A77.

17. Gunaydin, M. and Zagermann, M.: The gauging of five-dimensional N = 2 Maxwell–Einsteinsupergravity theories coupled to tensor multiplets, Nuclear Phys. B 572 (2000), 1–2, 131.


319

Long-Time Asymptotics of Solutions to the CauchyProblem for the Defocusing Nonlinear SchrödingerEquation with Finite-Density Initial Data.II. Dark Solitons on Continua

A. H. VARTANIANDepartment of Mathematics, Winthrop University, Rock Hill, SC 29733, U.S.A.e-mail: [email protected]

(Received: 8 January 2002; in final form: 7 August 2002)

Abstract. For Lax-pair isospectral deformations whose associated spectrum, for given initial data,consists of the disjoint union of a finitely denumerable discrete spectrum (solitons) and a continuousspectrum (continuum), the matrix Riemann–Hilbert problem approach is used to derive the leading-order asymptotics as |t | → ∞ (x/t ∼ O(1)) of solutions (u = u(x, t)) to the Cauchy problem forthe defocusing nonlinear Schrödinger equation (Df NLSE), i∂tu + ∂2

xu − 2(|u|2 − 1)u = 0, withfinite-density initial data

u(x, 0) =x→±∞ exp

(i(1 ∓ 1)θ

2

)(1 + o(1)), θ ∈ [0, 2π).

The Df NLSE dark soliton position shifts in the presence of the continuum are also obtained.

Mathematics Subject Classifications (2000): Primary: 35Q15, 37K40, 35Q55, 37K15; secondary:30E20, 30E25, 81U40.

Key words: asymptotics, direct and inverse scattering, reflection coefficient, Riemann–Hilbert prob-lems, singular integral equations.

1. Introduction

In direct detection systems making use of polarisation-preserving single-mode(PPSM) optical fibres, return-to-zero bright soliton (strictly speaking, soliton-like)pulses, which propagate in the anomalous group velocity dispersion (GVD) regime(wavelengths >1.3 µm in standard telecommunications fibres), have been shown tobe effective toward the partial resolution of the deleterious problem of performancedegradation caused by, for example, dispersive pulse spreading [1]. For coherentcommunications systems, nonreturn-to-zero dark soliton pulses, which propagatein the normal GVD regime (wavelengths <1.3 µm) and consist of a rapid dip inthe intensity of a broad pulse of a continuous wave background, offer an analogousbenefit [2–4].

320 A. H. VARTANIAN

A model for dark soliton pulse propagation in PPSM optical fibres in the pi-cosecond time scale, which describes the slowly varying amplitude of the com-plex field envelope, u = u(x, t), in normalised and dimensionless form, is theCauchy problem for the defocusing nonlinear Schrödinger equation (Df NLSE)with finite-density, or nonvanishing, initial data [1–4],

i∂tu+ ∂2xu− 2(|u|2 − 1)u = 0, (x, t) ∈ R× R,

u(x, 0) := u0(x) =x→±∞ exp

(i(1 ∓ 1)θ

2

)(1+ o(1)),

(1)

where u0(x) ∈ C∞(R), θ ∈ [0, 2π) (see Equation (3)), and o(1) is to be understoodin the sense that,

∀(k, l) ∈ Z�0 × Z�0, |x|k(

d

dx

)l(u0(x)− exp

(i(1∓ 1)θ

2

))=

x→±∞ 0.

It is shown in [5] that, for initial data satisfying

|x|k(

d

dx

)l(u0(x)− exp

(i(1∓ 1)θ

2

))=

x→±∞ 0, (k, l) ∈ Z�0 × Z�0,

the closure of the set of soliton, or reflectionless, potentials of the Df NLSE in thetopology of uniform convergence of functions on compact sets of R remains aninvariant set of the model ∀t ∈ R and not just for t = 0 (see, also, [6]).

When (temporal) dark solitons are launched sufficiently close together in opticalfibres, they interact not only through soliton-soliton interactions, but also throughsoliton-radiation-tail interactions. Such interactions manifest as a jitter in the ar-rival times of dark solitons, potentially resulting in their shift outside of somepredetermined timing window and giving rise to errors in the detected informa-tion [4]. Physically, the optical pulse adjusts its width as it propagates along theoptical fibre to evolve into a (multi-) dark soliton pulse/mode, and a part, howeversmall, of the pulse energy is shed in the form of an asymptotically decaying dis-persive wavetrain, manifesting as a low-level broadband background radiation (acontinuum of linear-like radiative waves/modes). Modulo an O(1) position shiftdue to cummulative interactions with other dark solitons and the (dispersive) con-tinuum, the dark soliton pulse/mode maintains its robust/stable properties. Fromthe physical and theoretical point of view, therefore, it is important to understandhow the dark solitons and continuum interact, and to be able to derive an explicitfunctional form for this process, namely, to study the asymptotics as |t| → ∞(x/t ∼ O(1)) of solutions to the Cauchy problem for the Df NLSE with finite-density initial data having a (not the only one possible) decomposition of the formu0(x) := usol(x) + urad(x), where u0(x) satisfies the conditions stated heretofore,usol(x) ‘generates’ the multi- or N-dark soliton component of the solution, andurad(x) is the ‘small’ nondark-soliton part giving rise to the dispersive compo-nent of the solution. In this paper, the leading- (O(1)) and next-to-leading-order

ASYMPTOTICS OF THE Df NLSE DARK SOLITONS ON CONTINUA 321

(O(|t|−1/2)) terms of the asymptotic expansion as |t| → ∞ (x/t ∼ O(1)) of thesolution to the Cauchy problem for the Df NLSE with finite-density initial dataare derived: they represent, respectively, the N-dark soliton component, and thedispersive continuum and nontrivial interaction/overlap of the N-dark solitons withthe continuum.

Within the framework of the inverse scattering method (ISM) [7–9] (see,also, [10]), it is well known that the Df NLSE is a completely integrable nonlin-ear evolution equation (NLEE) having a representation as an infinite-dimensionalHamiltonian system [11, 12]. Even though the analysis of completely integrableNLEEs with rapidly decaying, e.g., Schwartz class, initial data on R have receivedthe vast majority of the attention within the ISM framework, there have been ahandful of works devoted exclusively to the direct and inverse scattering analysisof completely integrable NLEEs belonging to the ZS-AKNS class with nonvanish-ing (as |x| → ∞) values of the initial data [13–15] (see, also, [16, 17]). Other,very interesting classes of finite-density-type initial data for completely integrableNLEEs have also been considered [18–32].

Within the ISM framework, the asymptotic analysis of the solution to the Cauchyproblem for the Df NLSE with finite-density initial data is divided into two steps:(1) the analysis of the solitonless (pure radiative, or continuous) component of thesolution; and (2) the inclusion of the N-dark soliton component via the applicationof a ‘dressing’ procedure to the solitonless background [33–37]. The complete de-tails of the asymptotic analysis that constitutes stage (1) of the two-step asymptoticparadigm above, which is quite technical and whose results are essential in orderto obtain those of the present paper, can be found in [38]: this paper addressesstage (2) of the above programme via the matrix Riemann–Hilbert problem (RHP)approach [7, 12, 39–47]. It is important to note that, to the best of the author’sknowledge as at the time of the presents, the first to obtain the asymptotics ofsolutions to the Df NLSE for finite-density initial data in the solitonless sector wereIts and Ustinov [48, 49].

This paper is organized as follows. In Section 2, the necessary facts from the di-rect and inverse scattering analysis for the Df NLSE with finite-density initial dataare given, the (matrix) RHP analysed asymptotically as |t| → ∞ (x/t ∼ O(1)) isstated, and the results of this paper are summarised in Theorems 2.2.1–2.2.4 (andCorollaries 2.2.1 and 2.2.2). In Section 3, an augmented RHP, which is equiva-lent to the original one stated in Section 2, is formulated, and it is shown that, ast → +∞, modulo exponentially small terms, the solution of the augmented RHPconverges to the solution of an explicitly solvable, model RHP. In Section 4, themodel RHP is solved asymptotically as t → +∞, from which the asymptotics ofu(x, t) and

∫ x

±∞(|u(x′, t)|2 − 1) dx′ are derived, and, in Appendix A, the – analo-gous – asymptotic analysis is succinctly reworked for the case when t → −∞. InAppendices B and C, respectively, formulae which are necessary in order to obtainthe remaining asymptotic results of this paper are presented, and a panoramic viewof the matrix RH theory in the L2-Sobolev space is given [44–46, 50].

322 A. H. VARTANIAN

2. The Riemann–Hilbert Problem and Summary of Results

In this section, a synopsis of the direct/inverse spectral analysis for Equation (1) isgiven, the matrix RHP studied asymptotically as |t| → ∞ (x/t ∼ O(1)) is stated,and the results of this paper are summarised in Theorems 2.2.1–2.2.4. Before doingso, however, it will be convenient to introduce the notation used throughout thiswork.

NOTATIONAL CONVENTIONS

(1) I = ( 1 00 1

)is the 2 × 2 identity matrix,

0 =(

0 00 0

),

σ1 =(

0 11 0

), σ2 =

(0 −ii 0

), and σ3 =

(1 00 −1

)are the Pauli matrices,

σ+ =(

0 10 0

)and σ− =

(0 01 0

)are, respectively, the raising and lowering matrices, sgn(x) := 0 if x = 0 andx|x|−1 if x �= 0, and R± := {x; ±x > 0};

(2) for a scalar ω and a 2 × 2 matrix ϒ , ωad(σ3)ϒ := ωσ3ϒω−σ3;(3) for each segment of an oriented contour D , according to the given orien-

tation, the ‘+’ side is to the left and the ‘−’ side is to the right as onetraverses the contour in the direction of orientation, that is, for a matrixAij (·), i, j ∈ {1, 2}, (Aij (·))± denote the nontangential limits

(Aij (z))± := limz′→z

z′∈± side of D

Aij (z′);

(4) for a matrix Aij (·), i, j ∈ {1, 2}, to have boundary values in the L2 sense onan oriented contour D , it is meant that

limz′→z

z′∈± side of D

∫D

|A(z′)− (A(z))±|2 |dz| = 0,

where |A(·)| denotes the Hilbert–Schmidt norm,

|A(·)| :=(

2∑i,j=1

Aij (·)Aij (·))1/2

,

with (•) denoting complex conjugation of (•);


(5) for 1 � p <∞ and D some point set,

Lp

M2(C)(D)

:={f : D →M2(C); ‖f (·)‖L

p

M2(C)(D) :=

(∫D

|f (z)|p |dz|)1/p

<∞},

and, for p = ∞,

L∞M2(C)(D)

:={g: D → M2(C); ‖g(·)‖L∞

M2(C)(D) := maxi,j∈{1,2}

supz∈D

|gij (z)| <∞};

(6) for D an unbounded domain of R, SC(D) (respectively, SM2(C)(D)) denotesthe Schwartz space on D, namely, the space of all infinitely continuously dif-ferentiable (smooth) C-valued (respectively, M2(C)-valued) functions whichtogether with all their derivatives tend to zero faster than any positive powerof | • |−1 as | • | → ∞, that is,

SC(D) := C∞(D) ∩{f : D → C; ‖f (·)‖k,l := sup

x∈R

∣∣∣∣xk( d

dx

)l

f (x)

∣∣∣∣<∞, (k, l) ∈ Z�0 × Z�0

}and

SM2(C)(D) := {F : D → M2(C); Fij (·) ∈ C∞(D), i, j ∈ {1, 2}}∩{G: D → M2(C); ‖Gij (·)‖k,l

:= maxi,j∈{1,2}

supx∈R

∣∣∣∣xk( d

dx

)l

Gij (x)

∣∣∣∣ <∞, (k, l) ∈

Z�0 × Z�0

}, and C∞

0 (∗) :=∞⋂k=0

Ck0(∗);

(7) for D an unbounded domain of R,

S1C(D) := SC(D) ∩ {h(z); ‖h(·)‖L∞(D) := sup

z∈D|h(z)| < 1};

(8) ‖F (·)‖∩p∈J Lp

M2(C)(∗) :=∑

p∈J ‖F (·)‖Lp

M2(C)(∗), with card(J ) <∞;

(9) for (µ, ν) ∈ R × R, the function (• − µ)iν: C \ (−∞, µ) → C: • �→eiν ln(•−µ), with the branch cut taken along (−∞, µ) and the principal branchof the logarithm chosen, ln(•−µ) := ln |• −µ|+ i arg(• − µ), arg(• − µ) ∈(−π, π);

(10) a contour, D , say, which is the finite union of piecewise-smooth, simple,closed curves, is said to be orientable if its complement, C\D , can always bedivided into two, possibly disconnected, disjoint open sets ✵+ and ✵−, eitherof which has finitely many components, such that D admits an orientation

324 A. H. VARTANIAN

so that it can either be viewed as a positively oriented boundary D+ for✵+ or as a negatively oriented boundary D− for ✵− [45], i.e., the (possiblydisconnected) components of C \ D can be coloured by + or − in such away that the + regions do not share boundary with the − regions, except,possibly, at finitely many points [46];

(11) for γ a nullhomologous path in a region D ⊂ C,

int(γ ) :={ζ ∈ D \ γ ; indγ (ζ ) := 1

2π i

∫γ

dζ ′

ζ ′ − ζ�= 0

}.

2.1. THE RHP FOR THE Df NLSE

In this subsection, the main results from the direct/inverse scattering analysis of theCauchy problem for the Df NLSE are succinctly recapitulated: since the proofs ofthese results are given in [38], only final results are stated.

PROPOSITION 2.1.1. The necessary and sufficient condition for the compatibilityof the following linear system (Lax pair), for arbitrary ζ ∈ C,

∂x$(x, t; ζ ) = U(x, t; ζ )$(x, t; ζ ),∂t$(x, t; ζ ) = V(x, t; ζ )$(x, t; ζ ), (2)

where

U(x, t; ζ ) = −iλ(ζ )σ3 +(

0 u

u 0

),

V(x, t; ζ ) = −2i(λ(ζ ))2σ3 + 2λ(ζ )

(0 u

u 0

)− i

(uu− 1 ∂xu

∂xu uu− 1

)σ3,

and λ(ζ ) = 12(ζ + ζ−1), with ∂∗ζ = 0, ∗ ∈ {x, t}, is that u = u(x, t) satisfies the

Df NLSE.

One proves Proposition 2.1.1 via the isospectral deformation condition(∂∗ζ = 0, ∗ ∈ {x, t}), and invoking the Frobenius compatibility condition,

∂t∂x$(x, t; ζ ) = ∂x∂t$(x, t; ζ ) ⇒ ∂tU(x, t; ζ ) − ∂xV(x, t; ζ )++ [U(x, t; ζ ),V(x, t; ζ )] = 0, ζ ∈ C,

where [A,B] := AB−BA is the matrix commutator (note that tr(U(x, t; ζ )) =tr(V(x, t; ζ )) = 0).

Remark 2.1.1. Note that, if u(x, t) is a solution of the Df NLSE with $(x, t; ζ )the corresponding solution of system (2), $(x, t; ζ )Q(ζ ), with Q(ζ ) ∈ M2(C),also solves system (2).

The ISM analysis for the Df NLSE is based on the direct scattering problem forthe (self-adjoint) operator (cf. Proposition 2.1.1)

OD := iσ3∂x −( 1

2(ζ + ζ−1) iu0(x)

iu0(x)12(ζ + ζ−1)

),


where u(x, 0) := u0(x) satisfies u0(x) =x→±∞ u0(±∞)(1+ o(1)), with

u0(±∞) := exp

(i(1 ∓ 1)θ

2

), θ ∈ [0, 2π)

(see Equation (3)),

u0(x) ∈ C∞(R), and u0(x)− u0(±∞) ∈ SC(R±).

DEFINITION 2.1.1. Let u(x, t) be the solution of the Df NLSE with

u(x, 0) := u0(x) =x→±∞ u0(±∞)(1+ o(1)),

where

u0(±∞) := exp

(i(1 ∓ 1)θ

2

), θ ∈ [0, 2π)

(see Equation (3)),

u0(x) ∈ C∞(R), and u0(x)− u0(±∞) ∈ SC(R±).

Define $±(x, 0; ζ ) as the (Jost) solutions of the first equation of system (2),OD$±(x, 0; ζ ) = 0, with the following asymptotics:

$±(x, 0; ζ ) =x→±∞

(e

i(1∓1)θ4 σ3

(1 −iζ−1

iζ−1 1

)+ o(1)

)e−ik(ζ )xσ3,

where k(ζ ) = 12(ζ − ζ−1).

COROLLARY 2.1.1. Let u(x, t) be the solution of the Cauchy problem for theDf NLSE and $(x, t; ζ ) the corresponding solution of system (2) with the asymp-totics stated in Definition 2.1.1. Then $(x, t; ζ ) satisfies the symmetry reductions

σ1$(x, t; ζ ) σ1 = $(x, t; ζ ) and $(x, t; ζ−1) = ζ$(x, t; ζ )σ2.

PROPOSITION 2.1.2. Set

$±(x, 0; ζ ) :=($±

11(ζ ) $±12(ζ )

$±21(ζ ) $±

22(ζ )

).

Then(

$+12(ζ )

$+22(ζ )

)and

($−

11(ζ )

$−21(ζ )

)have analytic continuation to C+ (respectively,(

$+11(ζ )

$+21(ζ )

)and

($−

12(ζ )

$−22(ζ )

)have analytic continuation to C−), the monodromy (scat-

tering) matrix, T(ζ ), is defined by

$−(x, 0; ζ ) := $+(x, 0; ζ )T(ζ ), Im(ζ ) = 0,

326 A. H. VARTANIAN

where

T(ζ ) =(a(ζ ) b(ζ )

b(ζ ) a(ζ )

),

with

a(ζ ) = (1− ζ−2)−1($+22(ζ )$

−11(ζ )− $+

12(ζ )$−21(ζ )

),

b(ζ ) = (1− ζ−2)−1($+22(ζ ) $

−21(ζ )− $+

12(ζ )$−11(ζ )

),

|a(ζ )|2 − |b(ζ )|2 = 1, a(ζ−1) = a(ζ ), b(ζ−1) = −b(ζ ),and

det($±(x, 0; ζ ))|ζ=±1 = 0.

COROLLARY 2.1.2. Let the reflection coefficient associated with the direct scat-tering problem for the operator OD be defined by r(ζ ) := b(ζ )/a(ζ ). Then

r(ζ−1) = −r(ζ ).Remark 2.1.2. Note that, even though a(ζ ) (respectively, a(ζ )) has an analytic

continuation off Im(ζ ) = 0 to C+ (respectively, C−) and is continuous on C+(respectively, C−), b(ζ ) does not, in general, have an analytic continuation to C\R.Furthermore, for the finite-density initial data considered here, it is shown in [14]that, using Volterra-type integral representations for the elements of $±(x, 0; ζ )and a successive approximations argument, r(ζ ) ∈ SC(R) (see, also, Part 1 of [12]).

LEMMA 2.1.1. Let u(x, t) be the solution of the Cauchy problem for the Df NLSEand $±(x, 0; ζ ) the corresponding (Jost) solutions of OD$±(x, 0; ζ ) = 0 givenin Definition 2.1.1. Then $±(x, 0; ζ ) have the following asymptotics:

$−(x, 0; ζ )=

ζ→∞ eiθ2 σ3

(I + 1

ζ

(i∫ x

−∞(|u0(x′)|2 − 1) dx′ −iu0(x)e−iθ

iu0(x) eiθ −i∫ x

−∞(|u0(x′)|2 − 1) dx′

)+

+O(ζ−2)

)e−ik(ζ )xσ3,

$+(x, 0; ζ )=

ζ→∞

(I + 1

ζ

(i∫ x

+∞(|u0(x′)|2 − 1) dx′ −iu0(x)

iu0(x) −i∫ x

+∞(|u0(x′)|2 − 1) dx′

)+

+O(ζ−2)

)e−ik(ζ )xσ3,

$−(x, 0; ζ ) =ζ→0

(ζ−1σ2e−iθ2 σ3 +O(1))e−ik(ζ )xσ3,

$+(x, 0; ζ ) =ζ→0

(ζ−1σ2 +O(1))e−ik(ζ )xσ3.


COROLLARY 2.1.3. The following asymptotics are valid:

a(ζ ) =ζ→∞ e

iθ2

(1+

(i∫ +∞

−∞(|u0(x

′)|2 − 1) dx′)ζ−1 +O(ζ−2)

),

a(ζ ) =ζ→0

e−iθ2 (1+O(ζ )),

r(ζ ) =ζ→∞ O(ζ−1), r(ζ ) =

ζ→0O(ζ );

in particular, r(0) = 0.

In [38] it is shown that, for u(x, 0) := u0(x) satisfying u0(x) =x→±∞ u0(±∞)×(1 + o(1)), with u0(±∞) := exp( i(1∓1)θ

2 ), θ ∈ [0, 2π) (see Equation (3)), u0(x) ∈C∞(R), and u0(x) − u0(±∞) ∈ SC(R±), σOD := spec(OD) = σd ∪ σc (σd ∩σc = ∅), where σd is the finitely denumerable ‘discrete’ spectrum given by σd =+a ∪ +a , where +a := {ςn; a(ζ )|ζ=ςn = 0, ςn = eiφn, φn ∈ (0, π), n ∈{1, 2, . . . , N}}, with

a(ζ ) = eiθ2

N∏n=1

(ζ − ςn)

(ζ − ςn)exp

(−∫ +∞

−∞ln(1− |r(µ)|2)

(µ− ζ )

dµ

2π i

), ζ ∈ C+,

0 � θ = −2N∑n=1

φn −∫ +∞

−∞ln(1 − |r(µ)|2)

µ

dµ

2π< 2π, (3)

and +a ∩ +a = ∅ (card(σd) = 2N), and σc is the ‘continuous’ spectrum givenby σc = {ζ ; Im(ζ ) = 0}, with orientation from −∞ to +∞ (card(σc) = ∞).Furthermore, it is shown in [38] that, for r(ζ ) ∈ S1

C(R) and |r(±1)| �= 1,

a(s + iε) =ε↓0

(−s)N exp(i(θ2 +

∑Nn=1 φn + P.V.

∫R\{s}

ln(1−|r(µ)|2)(µ−s)

dµ2π

))(1 − |r(s)|2)κsgn(s)

×× (1+ o(1)), s ∈ {±1},

where P.V.∫

denotes the principal value integral, with κ± real, possibly zero,constants, and (trace identity)∫ +∞

−∞(|u(x′, t)|2 − 1) dx′ = −2

N∑n=1

sin(φn)−∫ +∞

−∞ln(1− |r(µ)|2) dµ

2π. (4)

The ‘inverse part’ of the ISM analysis is invoked by re-introducing the t-de-pendence, namely, studying the ∂t$(x, t; ζ ) = V(x, t; ζ )$(x, t; ζ ) componentof system (2). The scattering map (S) u0(x) �→ r(ζ ) = R(u0(·)), which isa bijection for u0(x) satisfying the finite-density initial conditions and r(ζ ) ∈S1

C(R), linearises the Df NLSE flow in the sense that, since a(ζ, t) = a(ζ ) is the

‘generator’ of the integrals of motion and b(ζ, t) = b(ζ ) exp(4ik(ζ )λ(ζ )t) [12],

328 A. H. VARTANIAN

r(ζ, t) := b(ζ, t)/a(ζ, t) evolves in the scattering data phase space according tothe rule r(ζ, t) = r(ζ ) exp(4ik(ζ )λ(ζ )t). Set [38]

2(x, t; ζ ) :=

$−11(x,t;ζ )a(ζ )

$+12(x, t; ζ )

$−21(x,t;ζ )a(ζ )

$+22(x, t; ζ )

, ζ ∈ C+,

$+11(x, t; ζ ) $−

12(x,t;ζ )a(ζ)

$+21(x, t; ζ ) $−

22(x,t;ζ )a(ζ)

, ζ ∈ C−,

with $±(x, t; ζ ) the solutions of system (2): 2(x, t; ζ ) has the asymptotics [38]

2(x, t; ζ )=

ζ→∞

(I + 1

ζ

(i∫ x

+∞(|u(x′, t)|2 − 1) dx′ −iu(x, t)iu(x, t) −i

∫ x

+∞(|u(x′, t)|2 − 1) dx′

)+

+O(ζ−2)

)e−ik(ζ )(x+2λ(ζ )t)σ3,

2(x, t; ζ ) =ζ→0

(ζ−1σ2 +O(1))e−ik(ζ )(x+2λ(ζ )t)σ3.

LEMMA 2.1.2 ([38]). Let u(x, t) be the solution of the Cauchy problem for theDf NLSE with finite-density initial data u(x, 0) := u0(x) =x→±∞ u0(±∞)(1 +o(1)), where

u0(±∞) := exp

(i(1 ∓ 1)θ

2

),

0 � θ = −2N∑n=1

sin(φn)−∫ +∞

−∞ln(1 − |r(µ)|2)

µ

dµ

2π< 2π,

u0(x) ∈ C∞(R),

and u0(x)− u0(±∞) ∈ SC(R±). Set

m(x, t; ζ ) := 2(x, t; ζ ) exp(ik(ζ )(x + 2λ(ζ )t)σ3).

Then: (1) the bounded discrete set σd is finite; (2) the poles of m(x, t; ζ ) are simple;(3) the first (respectively, second) column of m(x, t; ζ ) has poles in C+ (respec-tively, C−) at {ςn}Nn=1 (respectively, {ςn}Nn=1); and (4) m(x, t; ζ ): C \ (σd ∪ σc)→M2(C) solves the following RHP:

(i) m(x, t; ζ ) is piecewise (sectionally) meromorphic ∀ζ ∈ C \ σc;(ii) m±(x, t; ζ ) := lim ζ ′→ζ

±Im(ζ ′)>0

m(x, t; ζ ′) satisfy the jump condition

m+(x, t; ζ ) = m−(x, t; ζ )G(x, t; ζ ), ζ ∈ R,


where G(x, t; ζ ) = exp(−ik(ζ )(x + 2λ(ζ )t)ad(σ3))( 1+r(ζ )r(ζ−1) r(ζ−1)

r(ζ ) 1

),

and r(ζ ), the reflection coefficient associated with the direct scattering prob-lem for the operator OD , satisfies r(ζ ) =ζ→0 O(ζ ), r(ζ ) =ζ→∞ O(ζ−1),

r(ζ−1) = −r(ζ ), and r(ζ ) ∈ S1C(R);

(iii) for the simple poles of m(x, t; ζ ) at {ςn}Nn=1 and {ςn}Nn=1, there exist nilpotentmatrices, with degree of nilpotency 2, such that m(x, t; ζ ) satisfies the polarconditions

Res(m(x, t; ζ ); ςn) = limζ→ςn

m(x, t; ζ )gn(x, t)σ−, n ∈ {1, 2, . . . , N},Res(m(x, t; ζ ); ςn) = σ1Res(m(x, t; ζ ); ςn) σ1, n ∈ {1, 2, . . . , N},

where gn(x, t) = gn exp(2ik(ςn)(x + 2λ(ςn)t)), with

gn := |γn|eiθγn (ςn − ςn) exp

(− iθ

2+∫ +∞

−∞ln(1 − |r(µ)|2)

(µ− ςn)

dµ

2π i

)×

×N∏k=1k �=n

(ςn − ςk

ςn − ςk

), θγn = ±π

2;

(iv) det(m(x, t; ζ ))|ζ=±1 = 0;(v) m(x, t; ζ ) =ζ→0 ζ

−1σ2 +O(1);(vi) m(x, t; ζ ) = ζ→∞

ζ∈C\(σd∪σc)I +O(ζ−1);

(vii) m(x, t; ζ ) possesses the symmetry reductions m(x, t; ζ ) = σ1m(x, t; ζ ) σ1

and m(x, t; ζ−1) = ζm(x, t; ζ )σ2.

For r(ζ ) ∈ S1C(R): (i) the RHP for m(x, t; ζ ) formulated above is uniquely as-

ymptotically solvable; and (ii) 2(x, t; ζ ) = m(x, t; ζ ) exp(−ik(ζ )(x+2λ(ζ )t)σ3)

solves system (2) with

u(x, t) := i limζ→∞

ζ∈C\(σd∪σc)(ζ(m(x, t; ζ ) − I))12 (5)

the solution of the Cauchy problem for the Df NLSE, and∫ x

+∞(|u(x′, t)|2 − 1) dx′ := −i lim

ζ→∞ζ∈C \(σd∪σc)

(ζ(m(x, t; ζ ) − I))11. (6)

Remark 2.1.3. In this paper, for r(ζ ) ∈ S1C(R), the solvability of the RHP for

m(x, t; ζ ) formulated in Lemma 2.1.2 is proved, via explicit construction, for allsufficiently large |t| (x/t ∼ O(1)): the solvability of the RHP in the solitonlesssector, σd ≡ ∅, for r(ζ ) ∈ S1

C(R), as |t| → ∞ and |x| → ∞ such that z0 := x/t ∼

O(1) and ∈ R \ {−2, 0, 2}, was proved in [38].

330 A. H. VARTANIAN

2.2. SUMMARY OF RESULTS

In this subsection, the results of this work are summarised in Theorems 2.2.1–2.2.4: before doing so, however, the following preamble is necessary. Recall fromSubsection 2.1 that ςn := eiφn , φn ∈ (0, π), n ∈ {1, 2, . . . , N}. Set ςn := ξn + iηn,where ξn = Re(ςn) = cos(φn) ∈ (−1, 1), and ηn = Im(ςn) = sin(φn) ∈ (0, 1).Throughout this paper, it is assumed that: (1) ξi �= ξj ∀i �= j ∈ {1, 2, . . . , N}; and(2) the following ordering (enumeration) for the elements of the discrete spectrum(solitons), σd , is taken, ξ1 > ξ2 > · · · > ξN .

Remark 2.2.1. Throughout this paper, the ‘symbols’ cS(♦), c(6, 7, 8), c(z1, z2,

z3, z4), c(•), and c, appearing in the various error estimates, are to be understood asfollows: (1) for ±♦ > 0, cS(♦) ∈ SC(R±); (2) for ±6 > 0, c(6, 7, 8) ∈ L∞

C(R± ×

C∗ × C∗), where C∗ := C \ {0}; (3) for (z1, z2) ∈ R± × R±, c(z1, z2, z3, z4) ∈L∞

C(R2± × C∗ × C∗); (4) for ±• > 0, c(•) ∈ L∞

C(D±), where D+ := (0, 2)

and D− := (−2, 0); and (5) c ∈ C∗. Even though the symbols cS(♦), c(6, 7, 8),c(z1, z2, z3, z4), c(•), and c are not, in general, equal, and should properly be de-noted as c1(·), c2(·), etc., the simplified notations cS(♦), c(6, 7, 8), c(z1, z2, z3, z4),c(•), and c are retained throughout in order to eschew a flood of superfluous nota-tion as well as to maintain consistency with the main theme of this work, namely, toderive explicitly the leading-order asymptotics and the classes to which the errorsbelong without regard to their precise z0-dependence.

Remark 2.2.2. In Theorems 2.2.1–2.2.4 below, one should keep, everywhere,the upper (respectively, lower) signs as t →+∞ (respectively, t → −∞).

THEOREM 2.2.1. For r(ζ ) ∈ S1C(R), let m(x, t; ζ ) be the solution of the Rie-

mann–Hilbert problem formulated in Lemma 2.1.2. Let u(x, t), the solution ofthe Cauchy problem for the Df NLSE with finite-density initial data u(x, 0) :=u0(x) =x→±∞ u0(±∞)(1 + o(1)), where

u0(±∞) := exp

(i(1 ∓ 1)θ

2

),

0 � θ = −2N∑n=1

sin(φn)−∫ +∞

−∞ln(1 − |r(µ)|2)

µ

dµ

2π< 2π,

u0(x) ∈ C∞(R),

and u0(x) − u0(±∞) ∈ SC(R±), be defined by Equation (5). Then, for θγm =εbπ/2, εb ∈ {±1}, m ∈ {1, 2, . . . , N}, as t → ±∞ and x → ∓∞ such thatz0 := x/t < −2 and (x, t) ∈ {(x, t); x + 2t cos(φm) = O(1), φm ∈ (0, π)},u(x, t)

= e−i(θ±(1)+s±)(uS(x, t)+

√ν(λ1)√|t|(λ1 − λ2) (z

20 + 32)1/4

(uC(x, t) + uSC(x, t))+

+O

((cS(λ1)c(λ2, λ3, λ4)√

λ1(z20 + 32)

+ cS(λ2)c(λ1, λ3, λ4)√λ2(z

20 + 32)

)ln|t|

(λ1 − λ2)t

)), (7)


where

θ+(j) =(∫ 0

−∞+∫ λ1

λ2

)ln(1 − |r(µ)|2)

µj

dµ

2π, j ∈ {0, 1}, (8)

θ−(l) =(∫ λ2

0+∫ +∞

λ1

)ln(1 − |r(µ)|2)

µl

dµ

2π, l ∈ {0, 1}, (9)

λ1 = − 12(a1 − (a2

1 − 4)1/2), λ2 = λ−11 ,

λ3 = − 12(a2 − i(4 − a2

2)1/2), λ4 = λ3,

a1 = 14(z0 − (z2

0 + 32)1/2), a2 = 14 (z0 + (z2

0 + 32)1/2),

(10)

0 < λ2 < λ1, |λ3|2 = 1, a1a2 = −2,

s+ = 2N∑

k=m+1

φk, s− = 2m−1∑k=1

φk, ν(z) = − 1

2πln(1− |r(z)|2), (11)

uS(x, t) = 1 + εbεP e−2iφm+9±(x,t)

(1 + εbεP e9±(x,t)), (12)

εP = sgn

((m−1∏k=1

sin( 12(φm + φk))

sin( 12(φm − φk))

)(N∏

k=m+1

sin( 12(φm + φk))


)−1)= (−1)N−m, (13)

9±(x, t) = −2 sin(φm)(x + 2t cos(φm)− x±m), (14)

x±m = ln(|γm|)2 sin(φm)

±N∑k=1

sgn(m− k)

2 sin(φm)ln

(∣∣∣∣sin( 12 (φm + φk))

sin( 12 (φm − φk))

∣∣∣∣)±± 1

2

(∫ λ2

0+∫ +∞

λ1

−∫ 0

−∞−∫ λ1

λ2

)ln(1− |r(µ)|2)

(µ2 − 2µ cos(φm)+ 1)

dµ

2π, (15)

uC(x, t) = ieis±(λ1e∓i(:±(z0,t )±(2∓1) π4 ) + λ2e±i(:±(z0,t )±(2∓1) π4 )), (16)

:±(z0, t) = ± arg r(λ1)± 4∑k∈J±

arg(λ1 − eiφk )−

− arg;(iν(λ1))± t (λ1 − λ2)(z0 + λ1 + λ2)++ ν(λ1) ln|t| + 3ν(λ1) ln(λ1 − λ2)++ 1

2ν(λ1) ln(z20 + 32)∓<±(λ1)± 1

2<±(0), (17)

332 A. H. VARTANIAN

<+(z) = 1

π

(∫ 0

−∞+∫ λ1

λ2

)ln|µ− z| d ln(1− |r(µ)|2), (18)

<−(z) = 1

π

(∫ λ2

0+∫ +∞

λ1

)ln|µ− z|d ln(1 − |r(µ)|2), (19)

∑k∈J+ :=

∑Nk=m+1,

∑k∈J− :=

∑m−1k=1 , ;(·) is the gamma function [51], and

uSC(x, t) =7∑

k=1

u(k)SC(x, t), (20)

with

u(1)SC(x, t) = −2iεbεP csc(φm) sin(s±) cos

(:±(z0, t)± (2∓ 1)

π

4

)×

× sinh(9±(x, t)),u(2)SC(x, t) = 2iεbεP (cos(φm)e

is±++ 2 sin(φm) sin(s±)) cos

(:±(z0, t)± (2∓ 1)

π

4

)e9

±(x,t),

u(3)SC(x, t) =

4iεbεPλ21 sin(φm) sin(s±)e9±(x,t)

(λ21 − 2λ1 cos(φm)+ 1)2

×

×(((λ1 + λ2) cos(φm)− 2) cos

(:±(z0, t)± (2∓ 1)

π

4

)±

± (λ1 − λ2) sin(φm) sin

(:±(z0, t)± (2 ∓ 1)

π

4

)),

u(4)SC(x, t) =

2iεbεPλ1 cos(φm)e9±(x,t)

(λ21 − 2λ1 cos(φm)+ 1)

×

×(

2 cos(s± − φm) cos

(:±(z0, t)± (2∓ 1)

π

4

)−

− (λ1 + λ2) cos(s±) cos

(:±(z0, t)± (2∓ 1)

π

4

)∓

∓ (λ1 − λ2) sin(s±) sin

(:±(z0, t)± (2∓ 1)

π

4

)),

u(5)SC(x, t) = −4εbεP sin(φm)e9

±(x,t)

(1− e29±(x,t))cos

(:±(z0, t)± (2 ∓ 1)

π

4

)×

× (e−is± + cos(s± − φm)e−iφm+29±(x,t)),

u(6)SC(x, t) =

4εP λ1 sin(φm)e9±(x,t)

(1 − e29±(x,t))(λ21 − 2λ1 cos(φm)+ 1)

×

×((e9

±(x,t)(1+ εbεP cos(φm)e−iφm+9±(x,t)))×


×(−2εP cos(s± − φm) cos

(:±(z0, t)± (2∓ 1)

π

4

)+

+ εP (λ1 + λ2) cos(s±) cos

(:±(z0, t)± (2∓ 1)

π

4

)±

± εP (λ1 − λ2) sin(s±) sin

(:±(z0, t)± (2∓ 1)

π

4

))+

+ (1 − εbεP e9±(x,t))×

×(

2iεb sin(s± − φm) cos

(:±(z0, t)± (2 ∓ 1)

π

4

)−

− iεb(λ1 + λ2) sin(s±) cos

(:±(z0, t)± (2∓ 1)

π

4

)±

± iεb(λ1 − λ2) cos(s±) sin

(:±(z0, t)± (2∓ 1)

π

4

))),

u(7)SC(x, t) = − 8λ2

1 sin2(φm)e−iφm+29±(x,t)

(1− e29±(x,t))(λ21 − 2λ1 cos(φm)+ 1)2

×

×(((λ1 + λ2) cos(φm)− 2) cos

(:±(z0, t)± (2∓ 1)

π

4

)±


(:±(z0, t)± (2∓ 1)

π

4

))×

×((1+ εbεP e9

±(x,t)) sin(s±)+ i

(1 − εbεP e9

±(x,t)

1 + εbεP e9±(x,t)

)cos(s±)

).

For the conditions stated in the formulation of the theorem, as t → ±∞ andx → ±∞ such that z0 > 2 and (x, t) ∈ {(x, t); x + 2t cos(φm) = O(1), φm ∈(−π, 0)},

u(x, t) = −e−i(ψ±(1)+s±)(uS(x, t) +

√ν(ℵ4)√|t|(ℵ3 − ℵ4) (z

20 + 32)1/4

(uC(x, t)+

+ uSC(x, t))+O

((cS(ℵ3)c(ℵ4,ℵ1,ℵ2)√

|ℵ3|(z20 + 32)

+ cS(ℵ4)c(ℵ3,ℵ1,ℵ2)√|ℵ4|(z2

0 + 32)

)×

× ln|t|(ℵ3 − ℵ4)t

)), (21)

where

ψ+(j) =(∫ ℵ4

−∞+∫ 0

ℵ3

)ln(1 − |r(µ)|2)

µj

dµ

2π, j ∈ {0, 1}, (22)

ψ−(l) =(∫ ℵ3

ℵ4

+∫ +∞

0

)ln(1 − |r(µ)|2)

µl

dµ

2π, l ∈ {0, 1}, (23)

334 A. H. VARTANIAN

ℵ1 = − 12(a1 − i(4 − a2

1)1/2), ℵ2 = ℵ1,

(24)ℵ3 = − 12(a2 − (a2

2 − 4)1/2), ℵ4 = ℵ−13 ,

ℵ4 < ℵ3 < 0, |ℵ1|2 = 1,

uS(x, t) = 1 + εbεP e−2iφm+✵±(x,t)

(1 + εbεP e✵±(x,t)), (25)

εP = sgn

((m−1∏k=1

(− sin( 12(φm + φk)))


)(N∏

k=m+1

(− sin( 12(φm + φk)))


)−1)= (−1)m−1, (26)

✵±(x, t) = −2 sin(φm)(x + 2t cos(φm)− x±m), (27)

x±m = ln(|γm|)2 sin(φm)

±N∑k=1

sgn(m− k)

2 sin(φm)ln

(∣∣∣∣sin( 12 (φm + φk))

sin( 12 (φm − φk))

∣∣∣∣)±± 1

2

(∫ ℵ4

−∞+∫ 0

ℵ3

−∫ ℵ3

ℵ4

−∫ +∞

0

)ln(1 − |r(µ)|2)

(µ2 + 2µ cos(φm)+ 1)

dµ

2π, (28)

uC(x, t) = ieis±(ℵ3e±i(2±(z0,t )±(2∓1) π4 ) + ℵ4e∓i(2±(z0,t )±(2∓1) π4 )), (29)

2±(z0, t) = ± arg r(ℵ4)± 4∑k∈J±

arg(ℵ4 + eiφk )− arg;(iν(ℵ4))±

± t (ℵ4 − ℵ3)(z0 + ℵ3 + ℵ4)+ ν(ℵ4) ln|t|++ 3ν(ℵ4) ln(ℵ3 − ℵ4)+ 1

2ν(ℵ4) ln(z20 + 32)∓>±(ℵ4)±

± 12>

±(0), (30)

>+(z) = 1

π

(∫ ℵ4

−∞+∫ 0

ℵ3

)ln|µ− z| d ln(1− |r(µ)|2), (31)

>−(z) = 1

π

(∫ ℵ3

ℵ4

+∫ +∞

0

)ln|µ− z| d ln(1 − |r(µ)|2), (32)

and

uSC(x, t) =7∑

k=1

u(k)SC(x, t), (33)


with

u(1)SC(x, t) = 2iεbεP csc(φm) sin(s±) cos

(2±(z0, t)± (2 ∓ 1)

π

4

)×

× sinh(✵±(x, t)),

u(2)SC(x, t) = −2iεbεP (cos(φm)e

is±++ 2 sin(φm) sin(s±)) cos

(2±(z0, t)± (2∓ 1)

π

4

)e✵±(x,t),

u(3)SC(x, t) =

4iεbεPℵ24 sin(φm) sin(s±)e✵±(x,t)

(ℵ24 + 2ℵ4 cos(φm)+ 1)2

×

×(((ℵ4 + ℵ3) cos(φm)+ 2) cos

(2±(z0, t)± (2 ∓ 1)

π

4

)±

± (ℵ4 − ℵ3) sin(φm) sin

(2±(z0, t)± (2∓ 1)

π

4

)),

u(4)SC(x, t) =

2iεbεPℵ4 cos(φm)e✵±(x,t)

(ℵ24 + 2ℵ4 cos(φm)+ 1)

×

×(


(2±(z0, t)± (2∓ 1)

π

4

)+

+ (ℵ4 + ℵ3) cos(s±) cos

(2±(z0, t)± (2 ∓ 1)

π

4

)±

± (ℵ4 − ℵ3) sin(s±) sin

(2±(z0, t)± (2 ∓ 1)

π

4

)),

u(5)SC(x, t) =

4εbεP sin(φm)e✵±(x,t)

(1 − e2✵±(x,t))cos

(2±(z0, t)± (2 ∓ 1)

π

4

)×

× (e−is± + cos(s± − φm)e−iφm+2✵±(x,t)),

u(6)SC(x, t) = − 4εPℵ4 sin(φm)e✵±(x,t)

(1− e2✵±(x,t))(ℵ24 + 2ℵ4 cos(φm)+ 1)

×

×((e✵±(x,t)(1+ εbεP cos(φm)e

−iφm+✵±(x,t)))×

×(

2εP cos(s± − φm) cos

(2±(z0, t)± (2∓ 1)

π

4

)+

+ εP (ℵ4 + ℵ3) cos(s±) cos

(2±(z0, t)± (2 ∓ 1)

π

4

)±

± εP (ℵ4 − ℵ3) sin(s±) sin

(2±(z0, t)± (2∓ 1)

π

4

))−

− (1− εbεP e✵±(x,t))×

336 A. H. VARTANIAN

×(

2iεb sin(s± − φm) cos

(2±(z0, t)± (2 ∓ 1)

π

4

)+

+ iεb(ℵ4 + ℵ3) sin(s±) cos

(2±(z0, t)± (2∓ 1)

π

4

)∓

∓ iεb(ℵ4 − ℵ3) cos(s±) sin

(2±(z0, t)± (2∓ 1)

π

4

))),

u(7)SC(x, t) = − 8ℵ2

4 sin2(φm)e−iφm+2✵±(x,t)

(1− e2✵±(x,t))(ℵ24 + 2ℵ4 cos(φm)+ 1)2

×

×(((ℵ4 + ℵ3) cos(φm)+ 2) cos

(2±(z0, t)± (2 ∓ 1)

π

4

)±


(2±(z0, t)± (2∓ 1)

π

4

))×

×((1+ εbεP e✵±(x,t)) sin(s±)+ i

(1− εbεP e✵±(x,t)

1+ εbεP e✵±(x,t)

)cos(s±)

).



u0(±∞) := exp

(i(1 ∓ 1)θ

2

),

0 � θ = −2N∑n=1

sin(φn)−∫ +∞

−∞ln(1 − |r(µ)|2)

µ

dµ

2π< 2π,

u0(x) ∈ C∞(R),

and u0(x)−u0(±∞) ∈ SC(R±), be defined by Equation (5), and∫ x

+∞(|u(x′, t)|2−1) dx′ be defined by Equation (6). Let ε ∈ {±1}. Then, for θγm = εbπ/2, εb ∈ {±1},m ∈ {1, 2, . . . , N}, as t → ±∞ and x → ∓∞ such that z0 < −2 and (x, t) ∈{(x, t); x + 2t cos(φm) = O(1), φm ∈ (0, π)},∫ x

sgn(ε)∞(|u(x′, t)|2 − 1) dx′

= S±ε + H±

ε + ES(x, t)++

√ν(λ1)√|t|(λ1 − λ2) (z

20 + 32)1/4

(EC(x, t) + ESC(x, t))+

+O

((cS(λ1)c(λ2, λ3, λ4)√

λ1(z20 + 32)

+ cS(λ2)c(λ1, λ3, λ4)√λ2(z

20 + 32)

)ln|t|

(λ1 − λ2)t

), (34)


where

S+ε =

{2∑N

k=m+1 sin(φk), ε = +1,

−2∑m

k=1 sin(φk), ε = −1,(35)

S−ε =

{2∑m−1

k=1 sin(φk), ε = +1,

−2∑N

k=m sin(φk), ε = −1,

H+ε =

{θ+(0), ε = +1,

−θ−(0), ε = −1,H−

ε ={θ−(0), ε = +1,

−θ+(0), ε = −1,(36)

ES(x, t) = 2εbεP sin(φm)e9±(x,t)

(1+ εbεP e9±(x,t)), (37)

EC(x, t) = −2 cos(s±) cos

(:±(z0, t)± (2 ∓ 1)

π

4

), (38)

and

ESC(x, t) =7∑

k=1

E (k)SC (x, t), (39)

with

E (1)SC (x, t) =

8λ21 sin2(φm) cos(s±)e29±(x,t)

(1 + εbεP e9±(x,t))2(λ21 − 2λ1 cos(φm)+ 1)2

×

×(((λ1 + λ2) cos(φm)− 2) cos

(:±(z0, t)± (2 ∓ 1)

π

4

)±


(:±(z0, t)± (2 ∓ 1)

π

4

)),

E (2)SC (x, t) =

4λ1εP sin(φm)e9±(x,t)

(1 − e29±(x,t))(λ21 − 2λ1 cos(φm)+ 1)

×

×(

2(εP cos(φm) sin(s± − φm)e9±(x,t) − εb sin(s±))×

× cos

(:±(z0, t)± (2 ∓ 1)

π

4

)−

− (λ1 + λ2)(εP cos(φm) sin(s±)e9±(x,t) − εb sin(s± + φm))×

× cos

(:±(z0, t)± (2 ∓ 1)

π

4

)±

± (λ1 − λ2)(εP cos(φm) cos(s±)e9±(x,t) − εb cos(s± + φm))×

× sin

(:±(z0, t)± (2 ∓ 1)

π

4

)),

338 A. H. VARTANIAN

E (3)SC (x, t) =

2εbεPλ1 cos(φm)e9±(x,t)

(λ21 − 2λ1 cos(φm)+ 1)

×

×(


(:±(z0, t)± (2∓ 1)

π

4

)−

− (λ1 + λ2) cos(s±) cos

(:±(z0, t)± (2 ∓ 1)

π

4

)∓

∓ (λ1 − λ2) sin(s±) sin

(:±(z0, t)± (2 ∓ 1)

π

4

)),

E (4)SC (x, t) =

4εbεPλ21 sin(φm) sin(s±)e9±(x,t)

(λ21 − 2λ1 cos(φm)+ 1)2

×

×(((λ1 + λ2) cos(φm)− 2) cos

(:±(z0, t)± (2 ∓ 1)

π

4

)±


(:±(z0, t)± (2 ∓ 1)

π

4

)),

E (5)SC (x, t) = −2εbεP csc(φm) sin(s±) cos

(:±(z0, t)± (2∓ 1)

π

4

)×

× sinh(9±(x, t)),

E (6)SC (x, t) =

4 sin(φm) sin(s± − φm)

(1− e29±(x,t))cos

(:±(z0, t)± (2 ∓ 1)

π

4

)e29±(x,t),

E (7)SC (x, t) = 2εbεP cos(s± − φm) cos

(:±(z0, t)± (2 ∓ 1)

π

4

)e9

±(x,t),

and θ±(·), {λn}4n=1, s± and ν(·), εP ,9±(x, t), and :±(z0, t) given in Theorem 2.2.1,

Equations (8)–(9), (10), (11), (13), (14)–(15), and (17)–(19), respectively.For the conditions stated in the formulation of the theorem, as t → ±∞ and

x → ±∞ such that z0 > 2 and (x, t) ∈ {(x, t); x + 2t cos(φm) = O(1), φm ∈(−π, 0)},

∫ x

sgn(ε)∞(|u(x′, t)|2 − 1) dx′

= S±ε + H±

ε + ES(x, t)++

√ν(ℵ4)√|t|(ℵ3 − ℵ4) (z

20 + 32)1/4

(EC(x, t) + ESC(x, t))+

+O

((cS(ℵ3)c(ℵ4,ℵ1,ℵ2)√

|ℵ3|(z20 + 32)

+ cS(ℵ4)c(ℵ3,ℵ1,ℵ2)√|ℵ4|(z2

0 + 32)

)ln|t|

(ℵ3 − ℵ4)t

),

(40)


where

S+ε =

{−2∑m

k=1 sin(φk), ε = +1,

2∑N

k=m+1 sin(φk), ε = −1,(41)

S−ε =

{−2∑N

k=m sin(φk), ε = +1,

2∑m−1

k=1 sin(φk), ε = −1,

H+ε =

{ψ−(0), ε = +1,

−ψ+(0), ε = −1,H−

ε ={ψ+(0), ε = +1,

−ψ−(0), ε = −1,(42)

ES(x, t) = 2εbεP sin(φm)e✵±(x,t)

(1+ εbεP e✵±(x,t)), (43)

EC(x, t) = 2 cos(s±) cos

(2±(z0, t)± (2∓ 1)

π

4

), (44)

and

ESC(x, t) =7∑

k=1

E (k)SC (x, t), (45)

with

E (1)SC (x, t) =

8ℵ24 sin2(φm) cos(s±)e2✵±(x,t)

(1 + εbεP e✵±(x,t))2(ℵ24 + 2ℵ4 cos(φm)+ 1)2

×

×(((ℵ4 + ℵ3) cos(φm)+ 2) cos

(2±(z0, t)± (2∓ 1)

π

4

)±


(2±(z0, t)± (2 ∓ 1)

π

4

)),

E (2)SC (x, t) =

4ℵ4εP sin(φm)e✵±(x,t)

(1 − e2✵±(x,t))(ℵ24 + 2ℵ4 cos(φm)+ 1)

×

×(

2(εP cos(φm) sin(s± − φm)e✵±(x,t) − εb sin(s±))×

× cos

(2±(z0, t)± (2∓ 1)

π

4

)+

+ (ℵ4 + ℵ3)(εP cos(φm) sin(s±)e✵±(x,t) − εb sin(s± + φm))×× cos

(2±(z0, t)± (2∓ 1)

π

4

)∓

∓ (ℵ4 − ℵ3)(εP cos(φm) cos(s±)e✵±(x,t) − εb cos(s± + φm))×× sin

(2±(z0, t)± (2∓ 1)

π

4

)),

340 A. H. VARTANIAN

E (3)SC (x, t) = −2εbεPℵ4 cos(φm)e✵±(x,t)

(ℵ24 + 2ℵ4 cos(φm)+ 1)

×

×(


(2±(z0, t)± (2 ∓ 1)

π

4

)+

+ (ℵ4 + ℵ3) cos(s±) cos

(2±(z0, t)± (2∓ 1)

π

4

)±

± (ℵ4 − ℵ3) sin(s±) sin

(2±(z0, t)± (2 ∓ 1)

π

4

)),

E (4)SC (x, t) = −4εbεPℵ2

4 sin(φm) sin(s±)e✵±(x,t)

(ℵ24 + 2ℵ4 cos(φm)+ 1)2

×

×(((ℵ4 + ℵ3) cos(φm)+ 2) cos

(2±(z0, t)± (2∓ 1)

π

4

)±


(2±(z0, t)± (2 ∓ 1)

π

4

)),

E (5)SC (x, t) = −2εbεP csc(φm) sin(s±) cos

(2±(z0, t)± (2∓ 1)

π

4

)×

× sinh(✵±(x, t)),

E (6)SC (x, t) = −4 sin(φm) sin(s± − φm)

(1− e2✵±(x,t))cos

(2±(z0, t)± (2 ∓ 1)

π

4

)e2✵±(x,t),

E (7)SC (x, t) = 2εbεP cos(s± − φm) cos

(2±(z0, t)± (2 ∓ 1)

π

4

)e✵±(x,t),

and ψ±(·), {ℵn}4n=1, εP , ✵±(x, t), and 2±(z0, t) given in Theorem 2.2.1, Equa-

tions (22)–(23), (24), (26), (27)–(28), and (30)–(32), respectively.

One important application of the asymptotic results obtained in this paper isrelated to the so-called N-dark soliton scattering, namely, the explicit calculationof the nth dark soliton position shift in the presence of the (nontrivial) continuousspectrum. Note that, unlike bright solitons of the focusing NLSE (with rapidlydecaying, in the sense of Schwartz, initial data), which undergo both position andphase shifts [7, 12, 52], dark solitons of the Df NLSE (for the finite-density initialdata considered here) only undergo a position shift [13]. This leads to the following(see, also, Corollary 2.2.2)

COROLLARY 2.2.1. Set

+xn := x+n − x−n and +xn := x+n − x−n , n ∈ {1, 2, . . . , N}.As t → ±∞ and x → ∓∞ such that z0 := x/t < −2 and (x, t) ∈ {(x, t); x +2t cos(φn) = O(1), φn ∈ (0, π)},


+xn =N∑k=1

sgn(n− k)

sin(φn)ln

(∣∣∣∣sin( 12 (φn + φk))

sin( 12 (φn − φk))

∣∣∣∣)++(∫ λ2

0+∫ +∞

λ1

−∫ 0

−∞−∫ λ1

λ2

)ln(1 − |r(µ)|2)

(µ2 − 2µ cos(φn)+ 1)

dµ

2π,

and, as t → ±∞ and x → ±∞ such that z0 > 2 and (x, t) ∈ {(x, t); x +2t cos(φn) = O(1), φn ∈ (−π, 0)},

+xn =N∑k=1

sgn(n− k)

sin(φn)ln

(∣∣∣∣sin( 12 (φn + φk))

sin( 12 (φn − φk))

∣∣∣∣)++(∫ ℵ4

−∞+∫ 0

ℵ3

−∫ ℵ3

ℵ4

−∫ +∞

0

)ln(1− |r(µ)|2)

(µ2 + 2µ cos(φn)+ 1)

dµ

2π.

Proof. Follows from the definition of +xn and +xn, and Theorem 2.2.1, Equa-tions (15) and (28). ✷THEOREM 2.2.3. For r(ζ ) ∈ S1

C(R), let m(x, t; ζ ) be the solution of the Rie-


u0(±∞) := exp

(i(1 ∓ 1)θ

2

),

0 � θ = −2N∑n=1

sin(φn)−∫ +∞

−∞ln(1 − |r(µ)|2)

µ

dµ

2π< 2π,

u0(x) ∈ C∞(R),

and u0(x) − u0(±∞) ∈ SC(R±), be defined by Equation (5). Then, for θγm =εbπ/2, εb ∈ {±1}, m ∈ {1, 2, . . . , N}, as t → ±∞ and x → ∓∞ such thatz0 := x/t ∈ (−2, 0) and (x, t) ∈ {(x, t); x + 2t cos(φm) = O(1), φm ∈ (0, π)},

u(x, t) = e−i(@±(1)+s±)((1+ εbεP e−2iφm+9±

8 (x,t))

(1 + εbεP e9±8 (x,t))

+

+O(e−4|t | min

k �=m∈{1,2,...,N}{sin(φk)| cos(φk)−cos(φm)|})), (46)

where s± and εP , respectively, are given in Theorem 2.2.1, Equations (11) and (13),

@+(j) =∫ 0

−∞ln(1 − |r(µ)|2)

µj

dµ

2π,

(47)

@−(j) =∫ +∞

0

ln(1 − |r(µ)|2)µj

dµ

2π, j ∈ {0, 1},

9±8 (x, t) = −2 sin(φm)(x + 2t cos(φm)− x±m,8), (48)

342 A. H. VARTANIAN

and

x±m,8 = ±N∑k=1

sgn(m− k)

2 sin(φm)ln

(∣∣∣∣sin( 12 (φm + φk))

sin( 12 (φm − φk))

∣∣∣∣)±± 1

2

(∫ +∞

0−∫ 0

−∞

)ln(1− |r(µ)|2)

(µ2 − 2µ cos(φm)+ 1)

dµ

2π+ ln(|γm|)

2 sin(φm), (49)

and, as t → ±∞ and x → ±∞ such that z0 ∈ (0, 2) and (x, t) ∈ {(x, t); x +2t cos(φm) = O(1), φm ∈ (−π, 0)},

u(x, t) = −e−i(@±(1)+s±)((1+ εbεP e−2iφm+9±

7 (x,t))

(1 + εbεP e9±7 (x,t))

+

+O(e−4|t |mink �=m∈{1,2,...,N}{| sin(φk)‖ cos(φk)−cos(φm)|})), (50)

where εP is given in Theorem 2.2.1, Equation (26),

9±7 (x, t) = −2 sin(φm)(x + 2t cos(φm)− x±m,7), (51)

and

x±m,7 = ±N∑k=1

sgn(m− k)

2 sin(φm)ln

(∣∣∣∣sin( 12 (φm + φk))

sin( 12 (φm − φk))

∣∣∣∣)±± 1

2

(∫ 0

−∞−∫ +∞

0

)ln(1− |r(µ)|2)

(µ2 + 2µ cos(φm)+ 1)

dµ

2π+ ln(|γm|)

2 sin(φm). (52)



u0(±∞) := exp

(i(1 ∓ 1)θ

2

),

0 � θ = −2N∑n=1

sin(φn)−∫ +∞

−∞ln(1 − |r(µ)|2)

µ

dµ

2π< 2π,

u0(x) ∈ C∞(R),

and u0(x)−u0(±∞) ∈ SC(R±), be defined by Equation (5), and∫ x

+∞(|u(x′, t)|2−1) dx′ be defined by Equation (6). Let ε ∈ {±1}. Then, for θγm = εbπ/2, εb ∈ {±1},m ∈ {1, 2, . . . , N}, as t → ±∞ and x → ∓∞ such that z0 ∈ (−2, 0) and


(x, t) ∈ {(x, t); x + 2t cos(φm) = O(1), φm ∈ (0, π)},∫ x

sgn(ε)∞(|u(x′, t)|2 − 1) dx′

= S±ε +H±

8,ε +2εbεP sin(φm)e

9±8 (x,t)

(1 + εbεP e9±8 (x,t))

+

+O(e−4|t | min

k �=m∈{1,2,...,N}{sin(φk)| cos(φk)−cos(φm)|}), (53)

where εP is given in Theorem 2.2.1, Equation (13), S±ε are given in Theorem 2.2.2,

Equation (35), 9±8 (x, t) are given in Theorem 2.2.3, Equations (48)–(49),

H+8,ε =

{@+(0), ε = +1,

−@−(0), ε = −1,H−

8,ε ={@−(0), ε = +1,

−@+(0), ε = −1,(54)

and @±(·) are given in Theorem 2.2.3, Equation (47), and, as t → ±∞ and x →±∞ such that z0 ∈ (0, 2) and (x, t) ∈ {(x, t); x + 2t cos(φm) = O(1), φm ∈(−π, 0)},∫ x

sgn(ε)∞(|u(x′, t)|2 − 1) dx′

= S±ε +H±

7,ε +2εbεP sin(φm)e

9±7 (x,t)

(1 + εbεP e9±7 (x,t))

+

+O(e−4|t | min

k �=m∈{1,2,...,N}{| sin(φk)‖ cos(φk)−cos(φm)|}), (55)

where εP is given in Theorem 2.2.1, Equation (26), S±ε are given in Theorem 2.2.2,

Equation (41), 9±7 (x, t) are given in Theorem 2.2.3, Equations (51)–(52), and

H+7,ε =

{@−(0), ε = +1,

−@+(0), ε = −1,H−

7,ε ={@+(0), ε = +1,

−@−(0), ε = −1.(56)

COROLLARY 2.2.2. Set

+x8n := x+n,8 − x−n,8 and +x7n := x+n,7 − x−n,7, n ∈ {1, 2, . . . , N}.As t → ±∞ and x → ∓∞ such that z0 := x/t ∈ (−2, 0) and (x, t) ∈ {(x, t); x+2t cos(φn) = O(1), φn ∈ (0, π)},

+x8n =N∑k=1

sgn(n− k)

sin(φn)ln

(∣∣∣∣sin( 12 (φn + φk))

sin( 12 (φn − φk))

∣∣∣∣)++(∫ +∞

0−∫ 0

−∞

)ln(1 − |r(µ)|2)

(µ2 − 2µ cos(φn)+ 1)

dµ

2π,

344 A. H. VARTANIAN

and, as t → ±∞ and x → ±∞ such that z0 ∈ (0, 2) and (x, t) ∈ {(x, t); x +2t cos(φn) = O(1), φn ∈ (−π, 0)},

+x7n =N∑k=1

sgn(n− k)

sin(φn)ln

(∣∣∣∣sin( 12 (φn + φk))

sin( 12 (φn − φk))

∣∣∣∣)++(∫ 0

−∞−∫ +∞

0

)ln(1 − |r(µ)|2)

(µ2 + 2µ cos(φn)+ 1)

dµ

2π.

Proof. Follows from the definition of +x8n and +x7n, and Theorem 2.2.3, Equa-tions (49) and (52). ✷

Remark 2.2.3. In this paper, the complete details of the asymptotic analysis arepresented for the case t → +∞ and x → −∞ such that z0 := x/t < −2 and(x, t) ∈ {(x, t); x + 2t cos(φn) = O(1), φn ∈ (0, π)}, and the final results for theanalogous asymptotic analysis as t → −∞ and x → +∞ such that z0 < −2 and(x, t) ∈ {(x, t); x + 2t cos(φn) = O(1), φn ∈ (0, π)} are given in Appendix A.The remaining cases are treated similarly, and one uses the results of Appendix Bto obtain the corresponding leading-order asymptotic expansions.

3. The Model RHP

In this section, the RHP studied asymptotically (as t → +∞) in Section 4, theso-called model RHP, is derived: it is obtained from the (normalised at ∞) RHP form(x, t; ζ ) formulated in Lemma 2.1.2 via an ingenius method due toDeift et al. [34] (see below). Set �m := {(x, t); x + 2t cos(φm) = O(1), φm ∈(0, π)}, m ∈ {1, 2, . . . , N}: note that the mth dark soliton ‘trajectory’ in the (x, t)-plane, R2, belongs to �m. From Lemma 2.1.2(iii), and the dark soliton orderingadopted in Subsection 2.2, one notes that, as t → +∞ and x → −∞ such thatz0 := x/t < −2 and (x, t) ∈ �m: (1) for n = m, gn(x, t) ��m

= O(1); (2)for n < m, gn(x, t) ��m

= O(exp(−4t sin(φn)| cos(φn) − cos(φm)|)) → 0; and(3) for n > m, gn(x, t) ��m

= O(exp(4t sin(φn)| cos(φn) − cos(φm)|)) → ∞.Thus (cf. Remark 2.1.3), since the RHP for m(x, t; ζ ) formulated in Lemma 2.1.2is asymptotically solvable for the (x, t)-sector stated above, one deduces that,along the trajectory of the (arbitrarily fixed) mth dark soliton: (1) for n = m,

Res(m(x, t; ζ ); ςn) =(

O(1) 0O(1) 0

)and Res(m(x, t; ζ ); ςn) =

( 0 O(1)0 O(1)

); (2) for n < m,

Res(m(x, t; ζ ); ςn) =(

O(�) 0O(�) 0

) → 0 and Res(m(x, t; ζ ); ςn) =( 0 O(�)

0 O(�)

) → 0,

where � := exp(−4t sin(φn)| cos(φn)−cos(φm)|); and (3) for n > m, Res(m(x, t;ζ ); ςn) =

(O(�−1) 0O(�−1) 0

)→ ( ∞ 0∞ 0

)and Res(m(x, t; ζ ); ςn) =

( 0 O(�−1)

0 O(�−1)

)→ ( 0 ∞0 ∞

).

Hence, along the trajectory of the (arbitrarily fixed) mth dark soliton, there are


exponentially growing polar (residue) conditions for solitons n with n ∈ {m +1,m + 2, . . . , N}. In a paper dealing with the Toda Rarefaction Problem [34],Deift et al. showed how this problem could be dealt with. Proceeding from theconstruction of Zhou [44–46] related to the singular RHP (see the synopsis belowTheorem C.1.4 in Appendix C), one uses the method of Deift et al. to ‘replace’ thepoles which give rise to the exponentially growing residue conditions by jump ma-trices on mutually disjoint, and disjoint with respect to σc, ‘small’ circles (see [46],Section 2, Remark 2.18, for a discussion about the radii of these circles) in sucha way that the jump matrices on these small circles behave like I + exponen-tially decreasing terms (as t → +∞), thus constructing the augmented contourσaugmented := σc ∪ (

⋃Nn=m+1 ∂(small circles)). Thus, instead of the original RHP,

one obtains an augmented (and normalised at ∞) RHP with 2(N −m) fewer polesand 2(N − m) additional circles with jump conditions stated on them. Finally, by‘removing’ the 2(N − m) small circles from the augmented RHP, one arrives atan asymptotically solvable, equivalent, ‘model’ RHP, in the sense that a solutionof the equivalent RHP gives a solution of the augmented RHP and vice versa; inparticular, if there are two RHPs, (X1(λ), υ1(λ), ;1) and (X2(λ), υ2(λ), ;2), say,with ;2 ⊂ ;1 and υ1(λ) �;1\;2=t→+∞ I+ o(1), then, modulo o(1) estimates, theirsolutions, X1(λ) and X2(λ), respectively, are asymptotically equal. Actually, aswill be shown below (see Lemma 3.5), the solution of the model RHP approxi-mates, up to terms that are exponentially small (as t → +∞), the solution of theaugmented RHP (hence the original RHP).

The reason for introducing the factor δ(ζ ) in Lemma 3.1 below is given inSection 4 of [38].

Remark 3.1. For notational convenience, all explicit x, t dependencies are here-after suppressed, except where absolutely necessary and/or where confusion mayarise.

LEMMA 3.1. For r(ζ ) ∈ S1C(R), let m(ζ): C\(σd∪σc)→ M2(C) be the solution

of the RHP formulated in Lemma 2.1.2. Set

m(ζ ) := m(ζ)(δ(ζ ))−σ3,

where

δ(ζ ) = exp

((∫ 0

−∞+∫ λ1

λ2

)ln(1− |r(µ)|2)

(µ− ζ )

dµ

2π i

),

with λ1 and λ2 given in Theorem 2.2.1, Equation (10), δ(ζ )δ(ζ ) = 1, δ(ζ )δ(ζ−1) =δ(0), and ‖(δ(·))±1‖L∞(C) := supζ∈C |(δ(ζ ))±1| <∞. Then m(ζ ): C\(σd∪σc)→M2(C) solves the following RHP:

(i) m(ζ ) is piecewise (sectionally) meromorphic ∀ζ ∈ C \ σc;

346 A. H. VARTANIAN

(ii) m±(ζ ) := lim ζ ′→ζ

±Im(ζ ′)>0m(ζ ′) satisfy the jump condition

m+(ζ ) = m−(ζ ) exp(−ik(ζ )(x + 2λ(ζ )t) ad(σ3))G(ζ ), ζ ∈ R,

where

G(ζ ) =((1− r(ζ )r(ζ ))δ−(ζ )(δ+(ζ ))−1 −r(ζ ) δ−(ζ )δ+(ζ )

r(ζ )(δ−(ζ )δ+(ζ ))−1 (δ−(ζ ))−1δ+(ζ )

);

(iii) m(ζ ) has simple poles in σd = ⋃Nn=1({ςn} ∪ {ςn}) with

Res(m(ζ ); ςn) = limζ→ςn

m(ζ )gn(δ(ςn))−2σ−, n ∈ {1, 2, . . . , N},

Res(m(ζ ); ςn) = σ1Res(m(ζ ); ςn) σ1, n ∈ {1, 2, . . . , N},where gn := |gn|eiθgn exp(2ik(ςn)(x + 2λ(ςn)t)), with

|gn| = 2|γn| sin(φn) exp

(∫ +∞

−∞sin(φn) ln(1 − |r(µ)|2)(µ2 − 2µ cos(φn)+ 1)

dµ

2π

)×

×N∏k=1k �=n

sin( 12(φn + φk))

sin( 12(φn − φk))

,

θgn = θγn +π

2− θ

2−∫ +∞

−∞(µ− cosφn) ln(1− |r(µ)|2)

(µ2 − 2µ cos(φn)+ 1)

dµ

2π−

−N∑k=1k �=n

φk, θγn = ±π

2;

(iv) det(m(ζ ))|ζ=±1 = 0;(v) m(ζ ) =ζ→0 ζ

−1(δ(0))σ3σ2 +O(1);(vi) m(ζ ) = ζ→∞


(vii) m(ζ ) = σ1m(ζ ) σ1 and m(ζ−1) = ζ m(ζ )(δ(0))σ3σ2.

Let


ζ∈C\(σd∪σc)(ζ(m(ζ )(δ(ζ ))σ3 − I))12, (57)

and ∫ x

+∞(|u(x′, t)|2 − 1) dx′ := −i lim

ζ→∞ζ∈C\(σd∪σc)

(ζ(m(ζ )(δ(ζ ))σ3 − I))11. (58)

Then u(x, t) is the solution of the Cauchy problem for the Df NLSE.Proof. The RHP for m(ζ ) (respectively, Equations (57) and (58)) follows from

the RHP for m(ζ) formulated in Lemma 2.1.2 (respectively, Equations (5) and (6))upon using m(ζ ) := m(ζ)(δ(ζ ))−σ3 , with δ(ζ ) given in the lemma. ✷


DEFINITION 3.1. For m ∈ {1, 2, . . . , N} and {ςn}Nn=m+1 ⊂ C+ (respectively,{ςn}Nn=m+1 ⊂ C−), define the clockwise (respectively, counter-clockwise) oriented

circles Kn := {ζ ; |ζ − ςn| = εKn } (respectively, Ln := {ζ ; |ζ − ςn| = εL

n }), withεKn (respectively, εL

n ) chosen sufficiently small such that Kn ∩ Kn′ = Ln ∩ Ln′ =Kn ∩ Ln = Kn ∩ σc = Ln ∩ σc = ∅ ∀n �= n′ ∈ {m+ 1,m+ 2, . . . , N}.

Remark 3.2. Note that the orientation for Kn (⊂ C+) and Ln (⊂ C−) is con-sistent with Equation (C.1) (see Appendix C).

LEMMA 3.2. For r(ζ ) ∈ S1C(R), let m(ζ ):C\(σd ∪σc)→ M2(C) be the solution

of the RHP formulated in Lemma 3.1. Set

m 6(ζ ) :=

m(ζ ),

ζ ∈ C \ (σc ∪ (⋃N

n=m+1(Kn ∪ int(Kn) ∪ Ln ∪ int(Ln)))),

m(ζ )(I − gn(δ(ςn))

−2

(ζ−ςn) σ−),

ζ ∈ int(Kn), n ∈ {m+ 1,m+ 2, . . . , N},m(ζ )

(I + gn(δ(ςn))−2

(ζ−ςn) σ+),

ζ ∈ int(Ln), n ∈ {m+ 1,m+ 2, . . . , N}.Then m 6(ζ ): C \ ((σd \⋃N

n=m+1({ςn} ∪ {ςn})) ∪ (σc ∪ (⋃N

n=m+1(Kn ∪ Ln))))→M2(C) solves the following RHP:

(i) m 6(ζ ) is piecewise (sectionally) meromorphic ∀ζ ∈ C\(σc∪(⋃Nn=m+1(Kn∪

Ln)));(ii) m

6±(ζ ) := lim ζ ′→ζ

ζ ′∈± side of σc∪(∪Nn=m+1(Kn∪Ln))

m 6(ζ ′) satisfy the jump condition

m6+(ζ ) = m

6−(ζ )υ6(ζ ), ζ ∈ σc ∪

(N⋃

n=m+1

(Kn ∪ Ln)

),

where

υ6(ζ ) =

exp(−ik(ζ )(x + 2λ(ζ )t) ad(σ3))G(ζ ), ζ ∈ R,

I + gn(δ(ςn))−2

(ζ−ςn) σ−, ζ ∈ Kn, n ∈ {m+ 1,m+ 2, . . . , N},I + gn(δ(ςn))−2

(ζ−ςn) σ+, ζ ∈ Ln, n ∈ {m+ 1,m+ 2, . . . , N},with G(ζ ) given in Lemma 3.1(ii);

(iii) m 6(ζ ) has simple poles in σd \⋃Nn=m+1({ςn} ∪ {ςn}) with

Res(m 6(ζ ); ςn) = limζ→ςn

m 6(ζ )gn(δ(ςn))−2σ−, n ∈ {1, 2, . . . , m},

Res(m 6(ζ ); ςn) = σ1Res(m 6(ζ ); ςn) σ1, n ∈ {1, 2, . . . , m};

(iv) det(m 6(ζ ))|ζ=±1 = 0;

348 A. H. VARTANIAN

(v) m 6(ζ ) =ζ→0 ζ−1(δ(0))σ3σ2 + O(1);

(vi) as ζ →∞, ζ ∈ C\((σd\⋃Nn=m+1({ςn}∪{ςn}))∪(σc∪(

⋃Nn=m+1(Kn∪Ln)))),

m6(ζ ) = I +O(ζ−1);(vii) m 6(ζ ) = σ1m

6(ζ ) σ1 and m6(ζ−1) = ζ m 6(ζ )(δ(0))σ3σ2.

For ζ ∈ C \ ((σd \⋃Nn=m+1({ςn} ∪ {ςn})) ∪ (σc ∪ (

⋃Nn=m+1(Kn ∪ Ln)))), let

u(x, t) := i limζ→∞(ζ(m

6(ζ )(δ(ζ ))σ3 − I))12, (59)

and ∫ x

+∞(|u(x′, t)|2 − 1) dx′ := −i lim

ζ→∞(ζ(m(ζ )(δ(ζ ))σ3 − I))11. (60)

Then u(x, t) is the solution of the Cauchy problem for the Df NLSE.Proof. The RHP for m 6(ζ ) (respectively, Equations (59) and (60)) follows from

the RHP for m(ζ ) formulated in Lemma 3.1 (respectively, Equations (57) and (58))upon using the definition of m 6(ζ ) in terms of m(ζ ) given in the lemma. ✷

Remark 3.3. Even though the set (of first-order poles)⋃N

n=m+1({ςn} ∪ {ςn}),giving rise to the exponentially growing residue conditions, has been removed fromthe specification of the RHP and replaced by jump matrices on

⋃Nn=m+1(Kn∪ Ln),

it should be noted that these jump matrices are also exponentially growing (ast → +∞). These lower/upper diagonal, exponentially growing jump matrices arenow replaced, via a finite sequence of transformations, by upper/lower diagonaljump matrices which converge to I as t →+∞.

LEMMA 3.3. For m ∈ {1, 2, . . . , N}, let σ ′d := σd \⋃n

n=m+1({ςn} ∪ {ςn}), σ ′c :=

σc ∪ (⋃N

n=m+1(Kn ∪ Ln)), where Kn and Ln are given in Definition 3.1, andσ ′

OD := σ ′d ∪ σ ′

c (σ′d ∩ σ ′

c = ∅). Set

m8(ζ ) :=

m 6(ζ )∏N

k=m+1(d+k (ζ ))

−σ3,

ζ ∈ C \ (σ ′c ∪ (

⋃Nn=m+1(int(Kn) ∪ int(Ln)))),

m 6(ζ )(JKn(ζ ))−1∏N

k=m+1(d−k (ζ ))

−σ3,

ζ ∈ int(Kn), n ∈ {m+ 1,m+ 2, . . . , N},m 6(ζ )(JLn

(ζ ))−1∏Nk=m+1(d

−k (ζ ))

−σ3,

ζ ∈ int(Ln), n ∈ {m+ 1,m+ 2, . . . , N},where

d+n (ζ ) =ζ − ςn

ζ − ςn, ζ ∈ C \

(σ ′c ∪(

N⋃n=m+1

(int(Kn) ∪ int(Ln))

)),

d−n (ζ ) ={ζ − ςn, ζ ∈ int(Kn),

(ζ − ςn)−1, ζ ∈ int(Ln),


JKn(ζ ) (∈ SL(2,C)) and JLn

(ζ ) (∈ SL(2,C)), respectively, are holomorphic in⋃Nk=m+1 int(Kk) and

⋃Nl=m+1 int(Ll), with

JKn(ζ )

=

∏Nk=m+1k �=n

d+k(ζ)

d−k(ζ)−CK

n gn(δ(ςn))−2

(ζ−ςn)2∏N

k=m+1k �=n

(d+k(ζ))−1

d−k(ζ)

(ζ−ςn)CKn

(ζ−ςn)2

∏Nk=m+1k �=n

(d+k (ζ ))−1

d−k (ζ )

−gn(δ(ςn))−2∏Nk=m+1k �=n

d−k (ζ )

d+k (ζ )(ζ − ςn)

∏Nk=m+1k �=n

d−k (ζ )d+k (ζ )

,

JLn(ζ )

= (ζ − ςn)

∏Nk=m+1k �=n

d+k (ζ )d−k (ζ )

gn(δ(ςn))−2∏N

k=m+1k �=n

d+k (ζ )d−k (ζ )

− CLn

(ζ−ςn)2

∏Nk=m+1k �=n

d−k (ζ )(d+k (ζ ))−1

∏Nk=m+1k �=n

d−k(ζ)

d+k(ζ)−CL

n gn(δ(ςn))−2

(ζ−ςn)2∏N

k=m+1k �=n

d−k(ζ)

(d+k(ζ))−1

(ζ−ςn)

,

and

CKn = CL

n = −4 sin2(φn)(gn)−1(δ(ςn))

2 e−2i

∑Nj=m+1j �=n

φj××

N∏k=m+1k �=n

(sin( 1

2 (φn + φk))

sin( 12 (φn − φk))

)2

.

Then m8(ζ ): C \ σ ′OD → M2(C) solves the following (augmented) RHP:

(i) m8(ζ ) is piecewise (sectionally) meromorphic ∀ζ ∈ C \ σ ′c;

(ii) m8±(ζ ) := lim ζ ′→ζ

ζ ′∈±side ofσ ′OD

m8(ζ ′) satisfy the following jump conditions,

m8+(ζ ) = m

8−(ζ ) exp(−ik(ζ )(x + 2λ(ζ )t) ad(σ3))G

8(ζ ), ζ ∈ R,

whereG8(ζ )

=(

(1− r(ζ )r(ζ ))δ−(ζ )(δ+(ζ ))−1 −r(ζ ) δ−(ζ )δ+(ζ )∏Nk=m+1(d

+k(ζ ))2

r(ζ )(δ−(ζ )δ+(ζ ))−1∏Nk=m+1(d

+k (ζ ))−2 (δ−(ζ ))−1δ+(ζ )

),

and

m8+(ζ ) =

m

8−(ζ )

(I + CK

n

(ζ−ςn)σ+),

ζ ∈ Kn, n ∈ {m+ 1,m+ 2, . . . , N},m

8−(ζ )

(I + CL

n

(ζ−ςn)σ−),

ζ ∈ Ln, n ∈ {m+ 1,m+ 2, . . . , N};(iii) m8(ζ ) has simple poles in σ ′

d with

Res(m8(ζ ); ςn) = limζ→ςn

m8(ζ )gn(δ(ςn))−2

(N∏

k=m+1

(d+k (ςn))−2

)σ−,

n ∈ {1, 2, . . . , m},Res(m8(ζ ); ςn) = σ1Res(m8(ζ ); ςn) σ1, n ∈ {1, 2, . . . , m};

350 A. H. VARTANIAN

(iv) det(m8(ζ ))|ζ=±1 = 0;(v) m8(ζ ) =ζ→0 ζ

−1(δ(0))σ3(∏N

k=m+1(d+k (0))

σ3)σ2 +O(1);(vi) m8(ζ ) = ζ→∞

ζ∈C \σ ′OD

I +O(ζ−1);

(vii) m8(ζ ) = σ1m8(ζ ) σ1 and m8(ζ−1) = ζ m8(ζ )(δ(0))σ3(∏N

k=m+1(d+k (0))

σ3)σ2.

Let


ζ∈C\σ ′OD

(ζ

(m8(ζ )(δ(ζ ))σ3

N∏k=m+1

(d+k (ζ ))σ3 − I

))12

, (61)

and ∫ x

+∞(|u(x′, t)|2 − 1) dx′

:= −i limζ→∞

ζ∈C\σ ′OD

(ζ

(m8(ζ )(δ(ζ ))σ3

N∏k=m+1

(d+k (ζ ))σ3 − I

))11

. (62)

Then u(x, t) is the solution of the Cauchy problem for the Df NLSE.Proof. From the definition of m8(ζ ) given in the lemma, one shows that, for

m ∈ {1, 2, . . . , N}, m8+(ζ ) = m

8−(ζ )υ

8

Kn(ζ ), ζ ∈ ⋃N

n=m+1 Kn, and m8+(ζ ) =

m8−(ζ )υ

8

Ln(ζ ), ζ ∈ ⋃N

n=m+1 Ln, where

υ8

Kn(ζ ) =

(N∏

k=m+1

(d−k (ζ ))σ3

)JKn

(ζ )

(I + gn(δ(ςn))

−2

(ζ − ςn)σ−) N∏

k=m+1

(d+k (ζ ))−σ3,

υ8

Ln(ζ ) =

(N∏

k=m+1

(d+k (ζ ))σ3

)(I+ gn(δ(ςn))

−2

(ζ − ςn)σ+)(JLn

(ζ ))−1N∏

k=m+1

(d−k (ζ ))−σ3 .

Now, as in [34], demanding that υ8

Kn(ζ ) (respectively, υ8

Ln(ζ )) have the following

upper (respectively, lower) diagonal structure, υ8

Kn(ζ ) = I + CK

n (ζ − ςn)−1σ+

(respectively, υ8

Ln(ζ ) = I + CL

n (ζ − ςn)−1σ−), one arrives at

JKn(ζ ) =∏N

k=m+1d+k (ζ )d−k (ζ )

− CKn gn(δ(ςn))

−2

(ζ−ςn)2

∏Nk=m+1

(d+k (ζ ))−1

d−k (ζ )

CKn

(ζ−ςn)∏N

k=m+1(d+k (ζ ))−1

d−k (ζ )

− gn(δ(ςn))−2

(ζ−ςn)∏N

k=m+1d−k (ζ )d+k (ζ )

∏Nk=m+1

d−k (ζ )d+k (ζ )

,

JLn(ζ ) = ∏N

k=m+1d+k (ζ )

d−k (ζ )

gn(δ(ςn))−2

(ζ−ςn)∏N

k=m+1d+k (ζ )d−k (ζ )

− CLn

(ζ−ςn)∏N

k=m+1d−k (ζ )

(d+k (ζ ))−1

∏Nk=m+1

d−k (ζ )

d+k (ζ )− CL

n gn(δ(ςn))−2

(ζ−ςn)2

∏Nk=m+1

d−k (ζ )(d+k (ζ ))−1

,


with det(JKn(ζ )) = det(JLn

(ζ )) = 1. Choosing d±n (ζ ) as in the lemma, one showsthat

Res(JKn(ζ ); ςn)

=((∏N

k=m+1k �=n

d+k (ζ )

d−k (ζ )− CK

n gn(δ(ςn))−2

(ζ−ςn)2

∏Nk=m+1k �=n

(d+k (ζ ))−1

d−k (ζ )

)∣∣∣ζ=ςn

0

0 0

),

Res(JLn(ζ ); ςn)

=( 0 0

0(∏N

k=m+1k �=n

d−k (ζ )d+k (ζ )

− CLn gn(δ(ςn))−2

(ζ−ςn)2

∏Nk=m+1k �=n

d−k (ζ )(d+k (ζ ))−1

)∣∣∣ζ=ςn

):

choosing CKn and CL

n as in the lemma, one gets that Res(JKn(ζ ); ςn) =

Res(JLn(ζ ); ςn) = 0; thus, JKn

(ζ ) (respectively, JLn(ζ )) is holomorphic in⋃N

n=m+1 int(Kn) (respectively,⋃N

n=m+1 int(Ln)). The remainder of the proof fol-lows from Lemma 3.2 and the definition of m8(ζ ) given in the lemma via straight-forward algebraic calculations. ✷

Remark 3.4. One notes from the proof of Lemma 3.3 that, for m ∈ {1, 2,. . . , N}, with ηn := sin(φn) ∈ (0, 1) and ξn := cos(φn) ∈ (−1, 1), as t → +∞and x →−∞ such that z0 := x/t < −2 and (x, t) ∈ �m,

υ8

Kn(ζ ) = I + CK

n

(ζ − ςn)σ+ = I +O

(e−4tηn|ξn−ξm|

(ζ − ςn)σ+),

ζ ∈ Kn, n ∈ {m+ 1,m+ 2, . . . , N},υ8

Ln(ζ ) = I + CL

n

(ζ − ςn)σ− = I +O

(e−4tηn|ξn−ξm|

(ζ − ςn)σ−),

ζ ∈ Ln, n ∈ {m+ 1,m+ 2, . . . , N};hence, as t → +∞, υ8

Dn(ζ ) → I (uniformly), where D ∈ {K,L}. One also notes

from Lemmae 3.1–3.3 that, for ζ ∈ ⋃Nn=m+1 int(Kn),

m8(ζ ) = m(ζ )

(ζ−ςnζ−ςn

)∏Nk=m+1k �=n

(d+k (ζ ))−1 − CK

n

(ζ−ςn)∏N

k=m+1k �=n

(d+k (ζ ))−1

0(ζ−ςnζ−ςn

)∏Nk=m+1k �=n

d+k (ζ )

,

and, for ζ ∈ ∪Nn=m+1int(Ln),

m8(ζ ) = m(ζ )

(ζ−ςnζ−ςn

)∏Nk=m+1k �=n

(d+k (ζ ))−1 0

CLn

(ζ−ςn)∏N

k=m+1k �=n

d+k (ζ )(ζ−ςnζ−ςn

)∏Nk=m+1k �=n

d+k (ζ )

;

hence, modulo singular terms like (ζ−ςn)−1 and (ζ−ςn)

−1, and recalling that (seeabove), as t → +∞, CK

n and CLn are O(exp(−4tηn|ξn − ξm|)), one deduces that,

352 A. H. VARTANIAN

since the RHP for m(ζ ) formulated in Lemma 3.1 is asymptotically solvable [38],there are no exponentially growing factors for m8(ζ ) when ζ ∈ ⋃N

n=m+1(int(Kn)∪int(Ln)).

By estimating the error along the trajectory of the mth dark soliton (m ∈ {1, 2,. . . , N}) when the jump matrices on {Kn, Ln}Nn=m+1 are removed from the spec-ification of the RHP for m8(ζ ), one arrives at an asymptotically solvable, modelRHP (see Lemma 3.5 below); however, since the proof of Lemma 3.5 relies sub-stantially on the Beals–Coifman (BC) construction [41] for the solution of a matrix(and appropriately normalised) RHP on an oriented and unbounded contour, it isconvenient to present, with some requisite preamble, a succinct and self-containedsynopsis of it at this juncture. But first, the following result is necessary.

PROPOSITION 3.1 ([38]). The solution of the RHP for m8(ζ ): C \ σ ′OD →

M2(C) formulated in Lemma 3.3 has the (integral equation) representation

m8(ζ ) = (I + ζ−1+8

0)P8(ζ )

(m

8

d(ζ )+∫σ ′c

m8−(µ)(υ8(µ)− I)

(µ− ζ )

dµ

2π i

),

ζ ∈ C \ σ ′OD ,

where

m8

d(ζ ) = I +m∑

n=1

(Res(m8(ζ ); ςn)

(ζ − ςn)+ σ1Res(m8(ζ ); ςn) σ1

(ζ − ςn)

),

v8(·) is a generic notation for the jump matrices of m8(ζ ) on σ ′c (Lemma 3.3(ii)),

and +8

0 and P 8(ζ ) are specified below. The solution of the above (integral) equa-tion can be written as the ordered factorisation

m8(ζ ) = (I + ζ−1+8

0)P8(ζ )m

8

d(ζ )mc(ζ ), ζ ∈ C \ σ ′

OD ,

where m8

d(ζ ) = σ1m8

d(ζ ) σ1 (∈ SL(2,C)) has the representation given above,

P 8(ζ ) = σ1P 8(ζ ) σ1 is chosen so that +8

0 is idempotent, I + ζ−1+8

0 (∈ M2(C))

is holomorphic in a punctured neighbourhood of the origin, with +8

0 = σ1+8

0 σ1

(∈ GL(2,C)) such that det(I + ζ−1+8

0)|ζ=±1 = 0, and having the finite, order 2,matrix involutive structure

+8

0 =(

+8ei(k+1/2)π (1 + (+8)2)1/2e−iϑ8

(1+ (+8)2)1/2eiϑ8

+8e−i(k+1/2)π

), k ∈ Z,

where +8 and ϑ 8 are obtained from the relation +8

0 = P 8(0)m8d(0)m

c(0)(δ(0))σ3×(∏N

k=m+1(d+k (0))

σ3)σ2, and satisfying tr(+8

0) = 0, det(+8

0) = −1, and +8

0+8

0 = I,and mc(ζ ): C \ σ ′

c → SL(2,C) solves the following RHP: (1) mc(ζ ) is piece-wise (sectionally) holomorphic ∀ζ ∈ C \ σ ′

c; (2) mc±(ζ ) := lim ζ ′→ζ

ζ ′∈± side of σ ′cmc(ζ ′)


satisfy, for ζ ∈ σ ′c, the jump condition mc+(ζ ) = mc−(ζ )υc(ζ ), where υc(ζ ) =

exp(−ik(ζ )(x + 2λ(ζ )t)ad(σ3))G8(ζ ), ζ ∈ R, with G8(ζ ) given in Lemma 3.3(ii),

υc(ζ ) = I + CKn (ζ − ςn)

−1σ+, ζ ∈ Kn, and υc(ζ ) = I + CLn (ζ − ςn)

−1σ−,ζ ∈ Ln, n ∈ {m + 1,m + 2, . . . , N}, with CK

n and CLn given in Lemma 3.3;

(3) mc(ζ ) = ζ→∞ζ∈C\σ ′c

I +O(ζ−1); and (4) mc(ζ ) = σ1mc(ζ ) σ1.

The BC formulation [41] now follows. One agrees to call a contour ;8 orientedif: (1) C\;8 has finitely many open connected components; (2) C\;8 is the disjointunion of two, possibly disconnected, open regions, denoted by ✵+ and ✵−; and(3) ;8 may be viewed as either the positively oriented boundary for ✵+ or the neg-atively oriented boundary for ✵− (C\;8 is coloured by two colours, ±). Let ;8, asa closed set, be the union of finitely many oriented simple piecewise-smooth arcs.Denote the set of all self-intersections of ;8 by ;8 (with card(;8) < ∞ assumedthroughout). Set ;8 := ;8 \ ;8. The BC construction for the solution of a (matrix)RHP, in the absence of a discrete spectrum and spectral singularities [45, 53], on anoriented contour ;8 consists of finding an M2(C)-valued function X(λ) such that:(1) X(λ) is piecewise holomorphic ∀λ ∈ C\;8; (2) X+(λ) = X−(λ)υ(λ), λ ∈ ;8,for some ‘jump’ matrix υ(λ): ;8 → GL(2,C); and (3) uniformly as λ → ∞,λ ∈ C \ ;8, X(λ) = I +O(λ−1). Let υ(λ) := (I −w−(λ))−1(I +w+(λ)), λ ∈ ;8,be a factorisation for υ(λ), where w±(λ) are some upper/lower, or lower/upper,triangular (depending on the orientation of ;8) nilpotent matrices, with degree ofnilpotency 2, and w±(λ) ∈⋂p∈{2,∞} L

p

M2(C)(;8) (if ;8 is unbounded, one requires

that w±(λ) = λ→∞λ∈;8

0). Define w(λ) := w+(λ)+ w−(λ), and introduce the Cauchy

operators on L2M2(C)(;

8),

(C±f )(λ) := limλ′→λ

λ′∈± side of;8

∫;8

f (z)

(z− λ′)dz

2π i,

where f (·) ∈ L2M2(C)(;

8), with C±: L2M2(C)(;

8) → L2M2(C)(;

8) bounded in op-erator normD, and ‖(C±f )(·)‖L2

M2(C)(∗) � const.||f (·)‖L2

M2(C)(∗). Introduce the BC

operator:

Cwf := C+(fw−)+ C−(fw+), f (·) ∈ L2M2(C)(∗);

moreover, since C\;8 can be coloured by two colours (±), C± are complementaryprojections [45], namely, C2+ = C+, C2− = −C−, C+C− = C−C+ = 0 (the nulloperator), and C+ − C− = id (the identity operator): in the case that C+ and−C− are complementary, the contour ;8 can always be oriented in such a waythat the ± regions lie on the ± sides of the contour, respectively. Specialising theBC construction to the solution of the RHP for mc(ζ ) on σ ′

c formulated in Propo-sition 3.1, and writing υc(ζ ) as the following (bounded) algebraic factorisation

D ‖C±‖N (;8) < ∞, where N (∗) denotes the space of all bounded linear operators acting from

L2M2(C)

(∗) into L2M2(C)

(∗).

354 A. H. VARTANIAN

υc(ζ ) := (I − wc−(ζ ))−1(I +wc+(ζ )), ζ ∈ σ ′c, the integral representation for mc(ζ )

is given by the following

LEMMA 3.4 (Beals and Coifman [41]). Let

µc(ζ ) = mc+(ζ )(I + wc

+(ζ ))−1 = mc

−(ζ )(I − wc−(ζ ))

−1, ζ ∈ σ ′c.

If µc(ζ ) ∈ I + L2M2(C)(σ

′c) := {I + h(·); h(·) ∈ L2

M2(C)(σ′c)}D solves the linear

singular integral equation

(id − Cwc)(µc(ζ )− I) = CwcI = C+(wc−)+ C−(wc

+), ζ ∈ σ ′c,

where id is the identity operator on L2M2(C)(σ

′c), then the solution of the RHP for

mc(ζ ) is

mc(ζ ) = I +∫σ ′c

µc(z)wc(z)

(z − ζ )

dz

2π i, ζ ∈ C \ σ ′

c,

where µc(ζ ) = ((id − Cwc)−1I)(ζ ), and wc(ζ ) := wc+(ζ )+ wc−(ζ ).

Finally, one arrives at, and is in a position to prove, the following

LEMMA 3.5. For m ∈ {1, 2, . . . , N}, set σd := ⋃mn=1({ςn} ∪ {ςn}), and let σc =

{ζ ; Im(ζ ) = 0} with orientation from −∞ to +∞. Let χ(ζ ): C \ (σd ∪ σc) →M2(C) solve the following RHP:

(i) χ(ζ ) is piecewise (sectionally) meromorphic ∀ζ ∈ C \ σc;(ii) χ±(ζ ) := lim ζ ′→ζ

ζ ′∈± side of σc

χ (ζ ′) satisfy the jump condition

χ+(ζ ) = χ−(ζ ) exp(−ik(ζ )(x + 2λ(ζ )t) ad(σ3))G8(ζ ), ζ ∈ R;

(iii) χ(ζ ) has simple poles in σd with

Res(χ(ζ ); ςn) = limζ→ςn

χ(ζ )gn(δ(ςn))−2

(N∏

k=m+1

(d+k (ςn))−2

)σ−,

n ∈ {1, 2, . . . , m},Res(χ(ζ ); ςn) = σ1Res(χ(ζ ); ςn) σ1, n ∈ {1, 2, . . . , m};

(iv) det(χ(ζ ))|ζ=±1 = 0;(v) χ(ζ ) =ζ→0 ζ

−1(δ(0))σ3(∏N

k=m+1(d+k (0))

σ3)σ2 +O(1);(vi) χ(ζ ) = ζ→∞


(vii) χ(ζ ) = σ1χ (ζ ) σ1 and χ (ζ−1) = ζ χ(ζ )(δ(0))σ3(∏N

k=m+1(d+k (0))

σ3)σ2.

D For f (ζ ) ∈ I + L2M2(C)

(∗), ‖f (·)‖I+L2M2(C)(∗) := (‖f (∞)‖2

L∞M2(C)(∗) + ‖f (·) −

f (∞)‖2L2

M2(C)(∗))1/2 [44].


Then, as t → +∞ and x → −∞ such that z0 := x/t < −2 and (x, t) ∈ �m,m8(ζ ): C \ σ ′

OD → M2(C) has the following asymptotics:

m8(ζ ) = (I +O(F (ζ ) exp(−� t)))χ(ζ ),

where � := 4 min m∈{1,2,...,N}n∈{m+1,m+2,...,N}

{sin(φn)| cos(φn)− cos(φm)|} (> 0), and, for i, j ∈{1, 2}, (F (ζ ))ij =ζ→∞ O(|ζ |−1) and (F (ζ ))ij =ζ→0 O(1). Furthermore, let


ζ∈C\(σd∪σc)

(ζ

(χ(ζ )(δ(ζ ))σ3

N∏k=m+1

(d+k (ζ ))σ3 − I

))12

+

+O(exp(−� t)), (63)

and ∫ x

+∞(|u(x′, t)|2 − 1) dx′

:= −i limζ→∞

ζ∈C\(σd∪σc)

(ζ

(χ(ζ )(δ(ζ ))σ3

N∏k=m+1

(d+k (ζ ))σ3 − I

))11

+

+O(exp(−� t)). (64)

Then u(x, t) is the solution of the Cauchy problem for the Df NLSE.

Remark 3.5. The solution of the (normalised at ∞) RHP for χ(ζ ): C \ (σd ∪σc) → M2(C) formulated in Lemma 3.5 has a factorised representation analogousto that of m8(ζ ) given in Proposition 3.1 (with appropriate change(s) of notation).

Proof. Define E(ζ ) := m8(ζ )(χ(ζ ))−1. From this definition, the RHPs form8(ζ ) and χ(ζ ) formulated in Lemmae 3.3 and 3.5, respectively, Proposition 3.1,and Remark 3.5, one shows that, for m ∈ {1, 2, . . . , N} and n ∈ {m + 1,m +2, . . . , N}, E(ζ ) solves the following RHP: (1) E(ζ ) is piecewise (sectionally)holomorphic ∀ζ ∈ C \ IE , where IE = ⋃N

n=m+1 InE , with In

E := Kn ∪ Ln

(with orientations preserved); (2) E±(ζ ) := lim ζ ′→ζ

ζ ′∈± side ofIE

E(ζ ′) satisfy the jump

condition E+(ζ ) = E−(ζ )υE(ζ ), ζ ∈ IE , where

υE (ζ ) ={

I + WKn

E (ζ ), ζ ∈ ⋃Nn=m+1 Kn (⊂ IE ), n ∈ {m+ 1,m+ 2, . . . , N},

I + WLn

E (ζ ), ζ ∈ ⋃Nn=m+1 Ln (⊂ IE), n ∈ {m+ 1,m+ 2, . . . , N},

with WKn

E (ζ ) = CKn (ζ − ςn)

−1X6(ζ ), WLn

E (ζ ) = CLn (ζ − ςn)

−1X7(ζ ),

X6(ζ ) =(−χ11(ζ )χ21(ζ ) (χ11(ζ ))

2

−(χ21(ζ ))2 χ11(ζ )χ21(ζ )

),

X7(ζ ) =(χ12(ζ )χ22(ζ ) −(χ12(ζ ))

2

(χ22(ζ ))2 −χ12(ζ )χ22(ζ )

),

356 A. H. VARTANIAN

and CKn and CL

n given in Lemma 3.3; (3) det(E(ζ ))|ζ=±1 = 1; (4) E(ζ ) =ζ→0 O(1)

and E = ζ→∞ζ∈C\IE

I + O(ζ−1); and (5) E(ζ ) = σ1E(ζ ) σ1 and E(ζ−1) = E(ζ ). Note,

in particular, that E(ζ ) has no jump discontinuity on R, and no poles. Recall, now,the BC construction (see the paragraph preceding Lemma 3.4). Write the following(bounded) algebraic factorisation for υE (ζ ), υE (ζ ) = (I − wE−(ζ ))−1(I + wE+(ζ )),ζ ∈ IE , and choose [46] wE−(ζ ) = 0; hence, wE+(ζ ) = WKn

E (ζ ), ζ ∈ ⋃Nn=m+1 Kn,

and wE+(ζ ) = WLn

E (ζ ), ζ ∈ ⋃Nn=m+1 Ln. Let µE (ζ ) be the solution of the BC

linear singular integral equation (idE − CwE )µE (ζ ) = I, ζ ∈ IE , where idE isthe identity operator on L2

M2(C)(IE ), and, for f (·) ∈ L2M2(C)(IE ), set CwEf :=

C+(fwE−)+ C−(fwE+) = C−(fwE+), with

(C±f )(ζ ) := limζ ′→ζ

ζ ′∈± side ofIE

∫IE

f (z)

(z− ζ ′)dz

2π i.

It was shown in [38] that ‖(idE − CwE )−1‖N (IE ) < ∞, where N (∗) denotes thespace of bounded linear operators from L2

M2(C)(∗) to L2M2(C)(∗). According to the

BC construction, the solution of the (normalised at ∞) RHP for E(ζ ) has theintegral representation

E(ζ ) = I +∫IE

µE (z)wE(z)

(z− ζ )

dz

2π i, ζ ∈ C \IE ,

where µE(ζ ) = ((idE −CwE )−1I)(ζ ), and wE(ζ ) =∑l∈{±}wEl (ζ ) = wE+(ζ ). Since

(cf. Definition 3.1), for i �= j ∈ {m+ 1,m + 2, . . . , N}, Ki ∩ Li = Ki ∩ Kj =Li ∩ Lj = ∅, it follows that

E(ζ ) = I +N∑

n=m+1

(∫Kn

µE (z)WKn

E (z)

(z − ζ )

dz

2π i+∫

Ln

µE (z)WLn

E (z)

(z− ζ )

dz

2π i

),

ζ ∈ C \IE .

From the second resolvent identity and the expressions for WKn

E (ζ ) and WLn

E (ζ ),one shows that

E(ζ )− I =N∑

n=m+1

(∫Kn

CKn X6(z)

(z − ςn)(z− ζ )

dz

2π i+∫

Ln

CLn X7(z)

(z− ςn)(z − ζ )

dz

2π i+

+∫

Kn

CKn ((idE − CwE )−1CwE I)(z)X6(z)

(z − ςn)(z− ζ )

dz

2π i+

+∫

Ln

CLn ((idE − CwE )−1CwE I)(z)X7(z)

(z− ςn)(z − ζ )

dz

2π i

), ζ ∈ C \IE .

Using the Cauchy–Schwarz inequality for integrals, one arrives at

|E(ζ )− I| �N∑

n=m+1

( |CKn |

2π‖X6(·)‖L2

M2(C)(Kn)

∥∥∥∥ I

(· − ςn)(· − ζ )

∥∥∥∥L2

M2(C)(Kn)

+


+ |CLn |

2π‖X7(·)‖L2

M2(C)(Ln)

∥∥∥∥ I

(· − ςn)(· − ζ )

∥∥∥∥L2

M2(C)(Ln)

+

+ |CKn |

2π‖(idE − CwE )−1‖N (Kn)

‖(CwE I)(·)‖L2M2(C)

(Kn)×

× ‖X6(·)‖L2M2(C)

(Kn)

∥∥∥∥ I

(· − ςn)(· − ζ )

∥∥∥∥L2

M2(C)(Kn)

+

+ |CLn |

2π‖(idE − CwE )−1‖N (Ln)

‖(CwE I)(·)‖L2M2(C)

(Ln)×

× ‖X7(·)‖L2M2(C)

(Ln)

∥∥∥∥ I

(· − ςn)(· − ζ )

∥∥∥∥L2

M2(C)(Ln)

), ζ ∈ C \IE .

One shows that, for ζ ∈ C \IE ,∥∥∥∥ I

(· − ςn)(· − ζ )

∥∥∥∥L2

M2(C)(Kn)

�√

2

εKn

(∫ 2π

0

dω

|ζ − εKn e−iω|2

)1/2

=:√

2

εKn

FKn(ζ ),

∥∥∥∥ I

(· − ςn)(· − ζ )

∥∥∥∥L2

M2(C)(Ln)

�√

2

εLn

(∫ 2π

0

dω

|ζ − εLn eiω|2

)1/2

=:√

2

εLn

FLn(ζ ),

with FDn (ζ ) =ζ→∞ O(|ζ |−1) and FDn(ζ ) =ζ→0 O(1), D ∈ {K,L}. Again, via theCauchy–Schwarz inequality for integrals,

‖(CwE I)(·)‖L2M2(C)(Kn)

� ‖(C−(IwE+))(·)‖L2

M2(C)(Kn)

� ‖C−‖N (Kn)‖wE

+(·)‖L2M2(C)

(Kn)

� ‖C−‖N (Kn)

∥∥∥∥ CKn

(· − ςn)X6(·)

∥∥∥∥L2

M2(C)(Kn)

� ‖C−‖N (Kn)|CK

n |‖X6(·)‖L2M2(C)

(Kn)

∥∥∥∥ I

(· − ςn)

∥∥∥∥L2

M2(C)(Kn)

� 2√

π

εKn

|CKn |‖C−‖N (Kn)

‖X6(·)‖L2M2(C)

(Kn),

with an analogous estimate for ‖(CwE I)(·)‖L2M2(C)(Ln)

:

‖(CwE I)(·)‖L2M2(C)(Ln)

� 2√

π

εLn

|CLn |‖C−‖N (Ln)

‖X7(·)‖L2M2(C)(Ln)

.

Hence, for ζ ∈ C \IE ,

|E(ζ )− I|

358 A. H. VARTANIAN

�N∑

n=m+1

( |CKn |FKn

(ζ )

π√

2εKn

‖X6(·)‖L2M2(C)

(Kn)+ |CL

n |FLn(ζ )

π√

2εLn

‖X7(·)‖L2M2(C)

(Ln)+

+√

2 |CKn |2FKn

(ζ )√π εK

n

‖(idE − CwE )−1‖N (Kn)‖C−‖N (Kn)

‖X6(·)‖2L2

M2(C)(Kn)

+

+√

2 |CLn |2FLn

(ζ )√π εL

n

‖(idE − CwE )−1‖N (Ln)‖C−‖N (Ln)

‖X7(·)‖2L2

M2(C)(Ln)

).

It is shown, a posteriori, in Section 4 that the RHP for χ (ζ ) formulated in theLemma is asymptotically solvable, whence ‖X6(·)‖2

L2M2(C)

(Kn)� const. = c and

‖X7(·)‖2L2

M2(C)(Ln)

� const. = c. Furthermore [38], ‖(idE − CwE )−1‖N (Dn) �const. ‖(idE − CwE )−1‖N (IE ) � c (see above), D ∈ {K,L}. Recalling the expres-sions for CK

n and CLn given in Lemma 3.3, that as t → +∞ and x → −∞ such

that z0 := x/t < −2 and (x, t) ∈ �m, (gn)−1 = O(exp(−4t sin(φn)|cos(φn) −cos(φm)|)), and the definition

‖E(·)− I‖L2M2(C)

(C \IE ):= max

i,j∈{1,2}sup

ζ∈C \IE

|(E(ζ )− I)ij |,

assembling the above, one arrives at

‖E(·)− I‖L2M2(C)

(C \IE )

� O(FE(ζ ) exp

(−4t min

m∈{1,2,...,N}n∈{m+1,m+2,...,N}

{sin(φn)|cos(φn)− cos(φm)|}))

,

where FE (ζ ) =ζ→∞ O(|ζ |−1) and FE(ζ ) =ζ→0 O(1); hence, the asymptotic esti-mate for m8(ζ ) stated in the lemma. Finally, from the asymptotics for E(ζ )− I de-rived above, the ordered factorisation for m8(ζ ) given in Proposition 3.1, and Equa-tions (61) and (62), the large-ζ asymptotics lead one to Equations (63) and (64). ✷

4. Asymptotic Solution of the Model RHP

In this section, the model (normalised at ∞) RHP for χ (ζ ) formulated in Lem-ma 3.5 is solved asymptotically as t → +∞ and x → −∞ such that z0 :=x/t < −2 and (x, t) ∈ �m, m ∈ {1, 2, . . . , N}, and the corresponding (asymp-totic) results for u(x, t), the solution of the Cauchy problem for the Df NLSE, and∫ x

±∞(|u(x′, t)|2 − 1) dx′ stated in Theorem 2.2.1 (for the upper sign) are derived.

LEMMA 4.1. The solution of the RHP for χ(ζ ): C \ (σd ∪ σc) → M2(C) formu-lated in Lemma 3.5 is given by the following ordered factorisation,

χ (ζ ) = (I + ζ−1+0)P (ζ )md(ζ )χc(ζ ), ζ ∈ C \ (σd ∪ σc),


where md(ζ ) = σ1md(ζ ) σ1 (∈ SL(2,C)) has the representation

md(ζ ) = I +m∑

n=1

(Res(χ(ζ ); ςn)

(ζ − ςn)+ σ1Res(χ(ζ ); ςn) σ1

(ζ − ςn)

),

P (ζ ) = σ1P (ζ ) σ1 is chosen (see Lemma 4.3 below) so that +0 is idempotent,I + ζ−1+0 is holomorphic in a punctured neighbourhood of the origin, with +0 =σ1+0 σ1 (∈ GL(2,C)) and det(I + ζ−1+0)|ζ=±1 = 0, and determined by therelation

+0 = P (0)md(0)χc(0)(δ(0))σ3

(N∏

k=m+1

(d+k (0))σ3

)σ2,

and satisfying tr(+0) = 0, det(+0) = −1, and +0+0 = I, and χc(ζ ): C \ σc →SL(2,C) solves the following RHP: (1) χc(ζ ) is piecewise (sectionally) holomor-phic ∀ζ ∈ C \ σc; (2) χc±(ζ ) := lim ζ ′→ζ

±Im(ζ ′)>0χc(ζ ′) satisfy, for ζ ∈ R, the jump

condition

χc+(ζ )= χc

−(ζ )e−ik(ζ )(x+2λ(ζ )t) ad(σ3)×

×((1 − r(ζ )r(ζ ))δ−(ζ )/δ+(ζ ) − r(ζ )

(δ−(ζ )δ+(ζ ))−1

∏Nk=m+1(d

+k (ζ ))

2

r(ζ )

δ−(ζ )δ+(ζ )∏N

k=m+1(d+k (ζ ))

−2 δ+(ζ )/δ−(ζ )

);

(3) χc(ζ ) = ζ→∞ζ∈C\σc

I +O(ζ−1); and (4) χc(ζ ) = σ1χc(ζ ) σ1.

Proof. One verifies that, modulo the explicit determination of +0, P (ζ ), md(ζ ),and χc(ζ ), the ordered factorisation for χ (ζ ) stated in the lemma, with the condi-tions on +0, P (ζ ), md(ζ ), and χc(ζ ) stated therein, solves the RHP for χ (ζ ) statedin Lemma 3.5. ✷

The determination of the asymptotics for the solution of the RHP for χc(ζ ): C\σc → SL(2,C) stated in Lemma 4.1 was the (principal) subject of study in [38],and is given by the following lemma:

LEMMA 4.2. Let ε be an arbitrarily fixed, sufficiently small positive real number,and, for z ∈ {λ1, λ2}, with λ1 and λ2 given in Theorem 2.2.1, Equation (10), setU(z; ε) := {ζ ; |ζ − z| < ε}. Then, as t → +∞ and x → −∞ such that z0 :=x/t < −2, for ζ ∈ C \⋃z∈{λ1,λ2} U(z; ε), χc(ζ ) has the following asymptotics:

χc11(ζ )

= 1 +O

((cS(λ1)c(λ2, λ3, λ3)√λ2(z

20 + 32) (ζ − λ1)

+ cS(λ2)c(λ1, λ3, λ3)√λ1(z

20 + 32) (ζ − λ2)

)ln t

(λ1 − λ2)t

),

360 A. H. VARTANIAN

χc12(ζ )

= ei<+(0)

2

( √ν(λ1) λ

2iν(λ1)1√

t (λ1 − λ2) (z20 + 32)1/4

(λ1e−i(:+(z0,t )+ π

4 )

(ζ − λ1)+ λ2ei(:+(z0,t )+ π

4 )

(ζ − λ2)

)+

+O

((cS(λ1)c(λ2, λ3, λ3)√λ2(z

20 + 32) (ζ − λ1)

+ cS(λ2)c(λ1, λ3, λ3)√λ1(z

20 + 32) (ζ − λ2)

)ln t

(λ1 − λ2)t

)),

χc21(ζ )

= e−i<+(0)

2

( √ν(λ1) λ

−2iν(λ1)

1√t (λ1 − λ2) (z

20 + 32)1/4

(λ1ei(:+(z0,t )+ π

4 )

(ζ − λ1)+ λ2e−i(:+(z0,t )+ π

4 )

(ζ − λ2)

)+

+O

((cS(λ1)c(λ2, λ3, λ3)√λ2(z

20 + 32) (ζ − λ1)

+ cS(λ2)c(λ1, λ3, λ3)√λ1(z

20 + 32) (ζ − λ2)

)ln t

(λ1 − λ2)t

)),

χc22(ζ )

= 1 +O

((cS(λ1)c(λ2, λ3, λ3)√λ2(z

20 + 32) (ζ − λ1)

+ cS(λ2)c(λ1, λ3, λ3)√λ1(z

20 + 32) (ζ − λ2)

)ln t

(λ1 − λ2)t

),

where λ3, ν(·), :+(z0, t), and <+(·), respectively, are given in Theorem 2.2.1,Equations (10), (11), (17), and (18), ‖(· − λk)

−1‖L∞(C \∪z∈{λ1,λ2}U(z;ε)) < ∞, k ∈{1, 2}, χc(ζ ) = σ1χ

c(ζ ) σ1, and (χc(0)σ2)2 = I (+O(t−1 ln t)).

Sketch of proof. Proceeding as in the proof of Lemma 6.1 in [38] and partic-ularising it to the case of the RHP for χc(ζ ) stated in Lemma 4.1, one arrivesat

χc11(ζ ) = 1− r(λ1)(δ

0B)

−2eπν2 e

iπ4

2π i(ζ − λ1)βIB0

21 XB

√t×

×∫ +∞

0

(e−

iπ4 ∂zD−iν(z)− i

2e

iπ4 zD−iν(z)

)z−iνe−

z24 dz+

+ r(λ1)(1− |r(λ1)|2)−1(δ0B)

−2e− 3π i4


21 e3πν

2 XB

√t

×

×∫ +∞

0

(e

3π i4 ∂zD−iν(z)− i

2e−

3π i4 zD−iν(z)

)z−iνe−

z24 dz−

− r(λ1)(δ0A)

−2e− πν2 (−1)iνe

3π i4

2π i(ζ − λ2)βIA0

21 XA

√t×

×∫ +∞

0

(e−

3π i4 ∂zDiν(z)+ i

2e

3π i4 zDiν(z)

)ziνe−

z24 dz+

+ r(λ1)(1 − |r(λ1)|2)−1(δ0A)

−2(−1)iνe− iπ4


21 eπν2 XA

√t

×


×∫ +∞

0

(e

iπ4 ∂zDiν(z)+ i

2e−

iπ4 zDiν(z)

)ziνe−

z24 dz+

+ O

((cS(λ1)c(λ2, λ3, λ3)(δ

0B)

−2

(ζ − λ1)|λ1 − λ3|√(λ1 − λ2) XB

+

+ cS(λ2)c(λ1, λ3, λ3)(δ0A)

−2

(ζ − λ2)|λ2 − λ3|√(λ1 − λ2)XA

)ln t

t

),

χc12(ζ ) =

(r(λ1)(δ

0B)

2eπν2 e− iπ

4

2π i(ζ − λ1)XB

√t− r(λ1)(1− |r(λ1)|2)−1(δ0

B)2e

3π i4

2π i(ζ − λ1)e3πν

2 XB

√t

)×

×∫ +∞

0Diν(z)z

iνe−z24 dz+

+(

r(λ1)(δ0A)

2e− πν2 e− 3π i

4

2π i(ζ − λ2)(−1)iνXA

√t− r(λ1)(1− |r(λ1)|2)−1(δ0

A)2e

iπ4

2π i(ζ − λ2)eπν2 (−1)iνXA

√t

)×

×∫ +∞

0D−iν(z)z

−iνe−z24 dz+

+ O

((cS(λ1)c(λ2, λ3, λ3)(δ

0B)

2

(ζ − λ1)|λ1 − λ3|√(λ1 − λ2)XB

+

+ cS(λ2)c(λ1, λ3, λ3)(δ0A)

2

(ζ − λ2)|λ2 − λ3|√(λ1 − λ2)XA

)ln t

t

),

χc21(ζ ) = −

(r(λ1)(δ

0B)

−2eπν2 e

iπ4

2π i(ζ − λ1)XB

√t− r(λ1)(1− |r(λ1)|2)−1(δ0

B)−2e− 3π i

4

2π i(ζ − λ1)e3πν

2 XB

√t

)×

×∫ +∞

0D−iν(z)z

−iνe−z24 dz−

−(

r(λ1)(δ0A)

−2e− πν2 e

3π i4

2π i(ζ − λ2)(−1)−iνXA

√t− r(λ1)(1 − |r(λ1)|2)−1(δ0

A)−2e− iπ

4

2π i(ζ − λ2)eπν2 (−1)−iνXA

√t

)×

×∫ +∞

0Diν(z)z

iνe−z24 dz+

+ O

((cS(λ1)c(λ2, λ3, λ3)(δ

0B)

−2

(ζ − λ1)|λ1 − λ3|√(λ1 − λ2)XB

+

+ cS(λ2)c(λ1, λ3, λ3)(δ0A)

−2

(ζ − λ2)|λ2 − λ3|√(λ1 − λ2)XA

)ln t

t

),

χc22(ζ ) = 1+ r(λ1)(δ

0B)

2eπν2 e− iπ

4


12 XB

√t×

×∫ +∞

0

(e

iπ4 ∂zDiν(z)+ i

2e−

iπ4 zDiν(z)

)ziνe−

z24 dz−

362 A. H. VARTANIAN

− r(λ1)(1 − |r(λ1)|2)−1(δ0B)

2e3π i4


12 e3πν

2 XB

√t×

×∫ +∞

0

(e−


2e

3π i4 zDiν(z)

)ziνe−

z24 dz+

+ r(λ1)(δ0A)

2e−πν2 e−

3π i4


12 (−1)iνXA

√t×

×∫ +∞

0

(e


2e−

3π i4 zD−iν(z)

)z−iνe−

z24 dz−

− r(λ1)(1− |r(λ1)|2)−1(δ0A)

2eiπ4


12 eπν2 (−1)iνXA

√t×

×∫ +∞

0

(e−


2e

iπ4 zD−iν(z)

)z−iνe−

z24 dz+

+ O

((cS(λ1)c(λ2, λ3, λ3)(δ

0B)

2

(ζ − λ1)|λ1 − λ3|√(λ1 − λ2)XB

+

+ cS(λ2)c(λ1, λ3, λ3)(δ0A)

2

(ζ − λ2)|λ2 − λ3|√(λ1 − λ2)XA

)ln t

t

),

where r(ζ ) = r(ζ )∏N

k=m+1(d+k (ζ ))

−2 (|r(λ1)| = |r(λ1)|), ν = ν(λ1),

δ0B = |λ1 − λ3|−iν(2t (λ1 − λ2)

3λ−31 )−

iν2 eZ(λ1)×

× exp

(− it

2(λ1 − λ2)(z0 + λ1 + λ2)

),

δ0A = |λ2 − λ3|iν(2t (λ1 − λ2)

3λ−32 )

iν2 eZ(λ2) exp

(it

2(λ1 − λ2)(z0 + λ1 + λ2)

),

Z(λ1) = i

2π

∫ 0

−∞ln|µ− λ1| d ln(1 − |r(µ)|2)+

+ i

2π

∫ λ1

λ2

ln|µ− λ1| d ln(1− |r(µ)|2),

Z(λ2) = −Z(λ1)+ i

2π

∫ 0

−∞ln|µ| d ln(1− |r(µ)|2)+

+ i

2π

∫ λ1

λ2

ln|µ| d ln(1 − |r(µ)|2),

XB = |λ1 − λ3|λ1

√2(λ1 − λ2)

λ1, XA = |λ2 − λ3|

λ2

√2(λ1 − λ2)

λ2,

βIB0

12 = βIB0

21 =√

2π e− πν2 e

iπ4

r(λ1) ;(iν), β

IA0

12 = βIA0

21 =√

2π e− πν2 e− iπ

4

r(λ1) ;(iν),


;(·) is the gamma function [51], and D∗(·) is the parabolic cylinder function [51].Using Equation (10), one shows that |λk − λ3|λ−1

k = (2λk)−1/2(z20 + 32)1/4, k ∈

{1, 2}. Using the identities [51] ∂zDz1(z) = 12 (z1Dz1−1(z)− Dz1+1(z)), zDz1(z) =

Dz1+1(z)+ z1Dz1−1(z), and |;(iν)|2 = π/(ν sinh(πν)), and the integral [51]∫ +∞

0D−z1(z)z

z2−1e−z2/4 dz

=√π exp(− 1

2(z1 + z2) ln 2);(z2)

;( 12(z1 + z2)+ 1

2 ), Re(z2) > 0,

from the above expressions for χcij (ζ ), i, j ∈ {1, 2}, and repeated application of

the relation |r(λ1)‖;(iν)|νeπν2 = (2πν)1/2, one obtains the result stated in the

lemma. Furthermore, one shows that the symmetry reduction χc(ζ ) = σ1χc(ζ ) σ1

is satisfied, and verifies that (χc(0)σ2)2 = I +O(t−1 ln t). ✷

PROPOSITION 4.1. For m ∈ {1, 2, . . . , N}, set Res(χ(ζ ); ςn) :=(an bncn dn

), n ∈

{1, 2, . . . , m}. Then bn = −anχc12(ςn)/χ

c22(ςn), dn = −cnχc

12(ςn)/χc22(ςn), and

{an, cn}mn=1 satisfy the following (nonsingular) system of 2m linear inhomogeneousalgebraic equations,

A B

B A

a1

a2...

amc1

c2...

cm

=

g∗1χc12(ς1)

g∗2χc12(ς2)...

g∗mχc12(ςm)

g∗1χc22(ς1)

g∗2χc22(ς2)...

g∗mχc22(ςm)

,

where

Aij :=

det(χc(ςi))+g∗i W(χc12(ςi),χ

c22(ςi))

χc22(ςi)

, i = j ∈ {1, 2, . . . , m},− g∗i (χc

12(ςi)χc22(ςj )−χc

22(ςi )χc12(ςj ))

(ςi−ςj )χc22(ςj )

, i �= j ∈ {1, 2, . . . , m},

Bij := −g∗i (χc22(ςi)χ

c22(ςj )− χc

12(ςi)χc12(ςj ))

(ςi − ςj)χc22(ςj )

, i, j ∈ {1, 2, . . . , m},

g∗j = |gj |eiθgj exp(2ik(ςj )(x + 2λ(ςj )t))(δ(ςj ))−2

N∏k=m+1

(d+k (ςj))−2,

j ∈ {1, 2, . . . , m},

364 A. H. VARTANIAN

with |gj | and θgj given in Lemma 3.1(iii), and

W(χc12(z), χ

c22(z)) =

∣∣∣∣ χc12(z) χc

22(z)

∂zχc12(z) ∂zχ

c22(z)

∣∣∣∣.Proof. Recall from Lemma 4.1 that χ(ζ ): C \ (σd ∪ σc) → M2(C) has the

factorised representation

χ (ζ ) = (I + ζ−1+0)P (ζ )××(

I +m∑

n=1

(Res(χ(ζ ); ςn)

(ζ − ςn)+ σ1Res(χ(ζ ); ςn) σ1

(ζ − ςn)

))χc(ζ ),

where χc(ζ ) is given in Lemma 4.2. For m ∈ {1, 2, . . . , N}, set Res(χ(ζ ); ςn) :=(an bncn dn

), whence σ1Res(χ(ζ ); ςn) σ1 =

(dn cnbn an

); thus,

χ(ζ ) =(

I + 1

ζ+0

)P (ζ )×

×(

1 + anζ−ςn +

∑mk=1k �=n

akζ−ςk +

∑mk=1

dkζ−ςk

bnζ−ςn +

∑mk=1k �=n

bkζ−ςk +

∑mk=1

ckζ−ςk

cnζ−ςn +

∑mk=1k �=n

ckζ−ςk +

∑mk=1

bkζ−ςk 1 + dn

ζ−ςn +∑m

k=1k �=n

dkζ−ςk +

∑mk=1

akζ−ςk

)×

×(χc

11(ζ ) χc12(ζ )

χc21(ζ ) χc

22(ζ )

). (65)

As in the BC construction [41], one now Taylor expands χc(ζ ) about {ςn}mn=1:χcij (ζ ) = χc

ij (ςn) + (∂ζχcij (ςn))(ζ − ςn) + O((ζ − ςn)

2), i, j ∈ {1, 2}, where∂ζχ

cij (ςn) = ∂ζχ

cij (ζ )|ζ=ςn . Recalling from Lemma 3.5(iii) that χ (ζ ) satisfies the

polar (residue) conditions Res(χ(ζ ); ςn) = limζ→ςn χ (ζ )g∗nσ− and Res(χ(ζ );

ςn) = σ1Res(χ(ζ ); ςn) σ1, n ∈ {1, 2, . . . , m}, with g∗n given in the proposition,assembling the above, one shows that the only nontrivial conditions are

anχc12(ςn)+ bnχ

c22(ςn) = 0,

cnχc12(ςn)+ dnχ

c22(ςn) = 0,

anχc11(ςn)+ bnχ

c21(ςn)

= ang∗n∂ζχ

c12(ςn)+

(1 +

m∑k=1k �=n

ak

ςn − ςk+

m∑k=1

dk

ςn − ςk

)g∗nχ

c12(ςn)+

+ bng∗n∂ζχ

c22(ςn)+

(m∑k=1k �=n

bk

ςn − ςk+

m∑k=1

ck

ςn − ςk

)g∗nχ

c22(ςn)+

+ limζ→ςn

( anχc12(ςn)+ bnχ

c22(ςn)︸︷︷︸

0

)g∗n

ζ − ςn,

cnχc11(ςn)+ dnχ

c21(ςn)


= cng∗n∂ζχ

c12(ςn)+

(m∑k=1k �=n

ck

ςn − ςk+

m∑k=1

bk

ςn − ςk

)g∗nχ

c12(ςn)+

+ dng∗n∂ζχ

c22(ςn)+

(1 +

m∑k=1k �=n

dk

ςn − ςk+

m∑k=1

ak

ςn − ςk

)g∗nχ

c22(ςn)+

+ limζ→ςn

( cnχc12(ςn)+ dnχ

c22(ςn)︸︷︷︸

0

)g∗n

ζ − ςn.

From the first two equations of the above system, one gets that bn = −anχc12(ςn)/

χc22(ςn) and dn = −cnχc

12(ςn)/χc22(ςn) (whence, det

(an bncn dn

) = 0): using the

latter (two) relations, it follows from the last two equations of the above systemthat, for n ∈ {1, 2, . . . , m},

anAn =m∑k=1k �=n

akg∗nBnk

ςn − ςk+

m∑k=1

ck g∗nDnk

ςn − ςk+ g∗nχ

c12(ςn),

cnAn =m∑k=1k �=n

ckg∗nBnk

ςn − ςk+

m∑k=1

akg∗nDnk

ςn − ςk+ g∗nχ

c22(ςn),

where

An = det(χc(ςn))+ g∗nW(χc12(ςn), χ

c22(ςn))

χc22(ςn)

,

Bnk = χc12(ςn)χ

c22(ςk)− χc

12(ςk)χc22(ςn)

χc22(ςk)

,

Dnk = χc22(ςn)χ

c22(ςk)− χc

12(ςn)χc12(ςk)

χc22(ςk)

;

thus, the (rank 2m) linear inhomogeneous algebraic system for {an, cn}mn=1 statedin the proposition. The nondegeneracy of the (2m × 2m) coefficient matrix isa consequence of the asymptotic solvability of the original RHP formulated inLemma 2.1.2 [38] (see, also, Equation (66) below). ✷PROPOSITION 4.2. As t → +∞ and x → −∞ such that z0 := x/t < −2 and(x, t) ∈ �m, m ∈ {1, 2, . . . , N}, for n ∈ {1, 2, . . . , m− 1},

an = O(e−ג+t ), bn = O(t−1/2(z20 + 32)−1/4e−ג+t ),

cn = O(e−ג+t ), dn = O(t−1/2(z20 + 32)−1/4e−ג+t ),

where +ג := 4 min m∈{1,2,...,N}n∈{1,2,...,m−1}

{sin(φn)|cos(φn)− cos(φm)|} (> 0), and

366 A. H. VARTANIAN

am = a0m +

1√ta1m +O

(cS(z0)

(z20 + 32)1/2

ln t

t

)=: g∗mg∗m (ςm − ςm)

−1

(1+ g∗mg∗m (ςm − ςm)−2)+

+ 1√t

(g∗mg∗m (ςm − ςm)

−1(g∗m∂ζ χc12(ςm)+ g∗m∂ζ χ

c12(ςm))

(1+ g∗mg∗m (ςm − ςm)−2)2+

+ g∗mχc12(ςm)

(1 + g∗mg∗m (ςm − ςm)−2)

)+O

(cS(z0)

(z20 + 32)1/2

ln t

t

),

bm = 1√tb1m +O

(cS(z0)

(z20 + 32)1/2

ln t

t

)=: − 1√

t

g∗mg∗m (ςm − ςm)−1χ c

12(ςm)

(1 + g∗mg∗m (ςm − ςm)−2)

+

+O

(cS(z0)

(z20 + 32)1/2

ln t

t

),

cm = c0m +

1√tc1m +O

(cS(z0)

(z20 + 32)1/2

ln t

t

)=: g∗m

(1+ g∗mg∗m (ςm − ςm)−2)+

+ 1√t

(g∗mg∗m (ςm − ςm)

−1χ c12(ςm)− g∗mg∗m∂ζ χ

c12(ςm)

(1 + g∗mg∗m (ςm − ςm)−2)+

+ g∗m(g∗m∂ζ χc12(ςm)+ g∗m∂ζ χ

c12(ςm))

(1 + g∗mg∗m (ςm − ςm)−2)2

)+O

(cS(z0)

(z20 + 32)1/2

ln t

t

),

dm = 1√td1m +O

(cS(z0)

(z20 + 32)1/2

ln t

t

)=: − 1√

t

g∗mχc12 (ςm)

(1+ g∗mg∗m (ςm − ςm)−2)

+

+O

(cS(z0)

(z20 + 32)1/2

ln t

t

),

where

χ c12(ζ ) =

√ν(λ1) e

i<+(0)2 λ

2iν(λ1)

1√(λ1 − λ2) (z

20 + 32)1/4

(λ1e−i(:+(z0,t )+ π

4 )

(ζ − λ1)+ λ2ei(:+(z0,t )+ π

4 )

(ζ − λ2)

),

with ν(·), λ1, λ2, λ3, <+(·), and :+(z0, t) specified in Lemma 4.2, and

cS(z0) = cS(λ1)c(λ2, λ3, λ3)√λ1 (λ1 − λ2)

+ cS(λ2)c(λ1, λ3, λ3)√λ2 (λ1 − λ2)

.

Proof. Noting that, as t → +∞ and x → −∞ such that z0 < −2 and (x, t) ∈�m, g∗n ��m

= O(1), n = m, and g∗n ��m= O(exp(−4t sin(φn)|cos(φn)−cos(φm)|)),


n ∈ {1, 2, . . . , m− 1}, one deduces from Proposition 4.1 that {an, cn}mn=1 solve

A1 o(1) . . . o(1)

o(1) A2. . .

......

. . .. . . o(1)

− g∗mBm1ςm−ς1 · · · − g∗mBmm−1

ςm−ςm−1Am

o(1) . . . . . . o(1)...

. . .. . .

......

. . .. . .

...

− g∗mDm1ςm−ς1 · · · · · · − g∗mDmm

ςm−ςm

o(1) . . . . . . o(1)...

. . .. . .

......

. . .. . .

...

− g∗mDm1ςm−ς1 · · · · · · − g∗mDmm

ςm−ςm

A1 o(1) . . . o(1)... A2

. . ....

.... . .

. . . o(1)

− g∗mBm1ςm−ς1 · · · − g∗mBmm−1

ςm−ςm−1Am

a1a2......

amc1c2......

cm

= [o(1), . . . , o(1), g∗mχ

c12(ςm)︸︷︷︸

m

, o(1), . . . , o(1), g∗mχc22(ςm)︸︷︷︸

m

]T,

where T denotes transposition, An, Bnk, and Dnk are given in the proof of Propo-sition 4.1, o(1) := O(exp(−4t min m∈{1,2,...,N}

n∈{1,2,...,m−1}{sin(φn)|cos(φn) − cos(φm)|})), and

χc12(·) and χc

22(·) are given in Lemma 4.2. Solving the above system for {an, cn}mn=1via the Cauchy–Binet formula, or Cramer’s rule, recalling the expressions for χc

ij (ζ ),i, j ∈ {1, 2}, given in Lemma 4.2, setting χ c

12(ζ ) and cS(z0) as in the proposi-tion, and recalling from Proposition 4.1 that bn = −anχc

12(ςn)/χc22(ςn) and dn =

−cnχc12(ςn)/χ

c22(ςn), one gets the estimates for {an, bn, cn, dn}m−1

n=1 and the explicit– asymptotic expansion – formulae for {am, bm, cm, dm} stated in the proposition.Furthermore, setting

Y := A B

B A

,

with A and B defined in Proposition 4.1, from the asymptotic estimates above for{an, bn, cn, dn}mn=1, and recalling that, as a consequence of the asymptotic solvabil-ity of the original RHP formulated in Lemma 2.1.2, det(Y) �≡ 0, an applicationof Hadamard’s inequality (|det(Y)|2 �

∏2mj=1

∑2mi=1 |Yij |2, where Yij denotes the

(i j)-element of Y) shows that

0 < |det(Y)|2 �m∏

j=1

(1 + sin2(φm)|γm|2P 2(φm, φk)Q

2(φm)

sin2( 12(φm + φj ))

e2φ(x,t)

)2

+

+ O

(cS(z0)

(z20 + 32)1/2

ln t

t

), (66)

where

φ(x, t) := −2 sin(φm)(x + 2t cosφm), (67)

368 A. H. VARTANIAN

P(φm, φk) :=(

m−1∏k=1

sin( 12 (φm + φk))

sin( 12 (φm − φk))

)(N∏

k=m+1

sin( 12 (φm + φk))

sin( 12 (φm − φk))

)−1

, (68)

Q(φm) := exp

((∫ λ2

0+∫ +∞

λ1

−∫ 0

−∞−∫ λ1

λ2

)×

× sin(φm) ln(1 − |r(µ)|2)(µ2 − 2µ cos(φm)+ 1)

dµ

2π

), (69)

and cS(z0) is given in the proposition. ✷The following lemma is proved via the higher-order generalisation [54] of the

Deift–Zhou (DZ) nonlinear steepest descent method [55] (see, also, [56]), but itsproof is far beyond the scope of the present work (it shall be presented elsewhere).

Remark 4.1. Even though in Lemma 4.3 below, in the sensus strictu of asymp-totic analysis, the exponentially small terms should be neglected, and thus not writ-ten out explicitly, in lieu of the t−p/2(ln t)q corrections, p � 1, q ∈ {0, 1, . . . , p −1}, they are written there, and there only (see, also, Appendix A, Lemma A.1.7),in order to bring to the reader’s attention the fact that there are additional, al-beit exponentially small, terms that are due to the remaining solitons: thereafter,exponentially small terms are neglected.

LEMMA 4.3. As t → +∞ and x → −∞ such that z0 := x/t < −2 and(x, t) ∈ �m, m ∈ {1, 2, . . . , N},

P (ζ ) = ζ+a+1

ζ+a+2a+3

ζ+a+4a+3

ζ+a+4ζ+a+1ζ+a+2

,

where

a+1 = a+2 = 1 +∞∑p=1

p−1∑q=0

a1pq(z0)(ln t)q

tp/2+

+O(e−4t min m∈{1,2,...,N}

n∈{1,2,...,m−1}{sin(φn)|cos(φn)−cos(φm)|})

,

a+3 =∞∑p=1

p−1∑q=0

a3pq(z0)(ln t)q

tp/2+O

(e−4t min m∈{1,2,...,N}


,

a+4 = 1+∞∑p=1

p−1∑q=0

a4pq(z0)(ln t)q

tp/2+O

(e−4t min m∈{1,2,...,N}


,

akpq(z0) ∈ cS(z0), k ∈ {1, 3, 4}, and P (ζ ) = σ1P (ζ ) σ1.


Remark 4.2. Even though Lemma 4.3 is not proven in this paper, it will beshown that (see the proof of Proposition 4.6 below), up to the leading-order termsretained in this work, namely, terms that are

O

(cS(z0)

(z20 + 32)1/2

ln t

t

), a3

10(z0) = 0;

thus, actually,

a+3 =∞∑p=2

p−1∑q=0

a3pq(z0)(ln t)q

tp/2+

+O(

exp(−4t min

m∈{1,2,...,N}n∈{1,2,...,m−1}

{sin(φn)|cos(φn)− cos(φm)|}))

= O

(cS(z0)

(z20 + 32)1/2

ln t

t

).

Furthermore, to

O

(cS(z0)

(z20 + 32)1/2

ln t

t

),

the asymptotic expansion for a+4 plays, in fact, no role in the final formulae ofthis paper. As a possible prelude to a motivation of why a+i , i ∈ {1, 2, 3, 4}, have,modulo exponentially small terms, the asymptotic expansions stated in Lemma 4.3,one can apply the higher-order generalisation of the DZ method [54] to the proof ofLemma 6.1 in [38] to show that χc

ij (ζ ), i, j ∈ {1, 2}, have the asymptotic expansion

χcij (ζ ) = δij +

∞∑p=1

p−1∑q=0

(χcij (z0))pq(fij (ζ ))pq(ln t)q

tp/2,

where δij is the Kronecker delta,

(χc11(·))10 = (χc

22(·))10 = 0, and ‖(fij (·))pq‖L∞(C\⋃z∈{λ1 ,λ2} U(z;ε)) <∞.

However, as stated heretofore, these details are omitted in this paper (it is theauthor’s conjecture that a+3 = a+4 = 0, namely, P (ζ ) is diagonal).

PROPOSITION 4.3. Set a110(z0) =: a1, a2

10(z0) =: a2, a310(z0) =: a3, and

a410(z0) =: a4. Then as t → +∞ and x → −∞ such that z0 := x/t < −2

and (x, t) ∈ �m, m ∈ {1, 2, . . . , N},

(+0)11 = − c0m

ςmiδ−1(0)e2i

∑Nk=m+1 φk+

+ iδ−1(0)e2i∑N

k=m+1 φk√t

(−(a1 − a2)

c0m

ςm−(b1m

ςm+ c1

m

ςm

)+

370 A. H. VARTANIAN

+ a3

(1 − a0

m

ςm

)−(

1 − a0m

ςm

)2δ(0)

√ν(λ1) cos(:+(z0, t)+ π

4 )√(λ1 − λ2) (z

20 + 32)1/4

)+

+O

(cS(z0)

(z20 + 32)1/2

ln t

t

),

(+0)12 = −(

1 − a0m

ςm

)iδ(0)e−2i

∑Nk=m+1 φk+

+ iδ(0)e−2i∑N

k=m+1 φk√t

(−(a1 − a2)

(1 − a0

m

ςm

)+(a1m

ςm+ d1

m

ςm

)+

+ a3c0m

ςm− c0

m

ςm

2δ−1(0)√ν(λ1) cos(:+(z0, t)+ π

4 )√(λ1 − λ2) (z

20 + 32)1/4

)+

+O

(cS(z0)

(z20 + 32)1/2

ln t

t

),

(+0)21 =(

1 − a0m

ςm

)iδ−1(0)e2i

∑Nk=m+1 φk+

+ iδ−1(0)e2i∑N

k=m+1 φk√t

((a1 − a2)

(1 − a0

m

ςm

)−(a1m

ςm+ d1

m

ςm

)−

− a3c0m

ςm+ c0

m

ςm

2δ(0)√ν(λ1) cos(:+(z0, t)+ π

4 )√(λ1 − λ2) (z

20 + 32)1/4

)+

+O

(cS(z0)

(z20 + 32)1/2

ln t

t

),

(+0)22 = c0m

ςmiδ(0)e−2i

∑Nk=m+1 φk+

+ iδ(0)e−2i∑N

k=m+1 φk√t

((a1 − a2)

c0m

ςm+(b1m

ςm+ c1

m

ςm

)− a3

(1 − a0

m

ςm

)+

+(

1 − a0m

ςm

)2δ−1(0)

√ν(λ1) cos(:+(z0, t)+ π

4 )√(λ1 − λ2) (z

20 + 32)1/4

)+

+O

(cS(z0)

(z20 + 32)1/2

ln t

t

).

Proof. Recall from Lemma 4.1 that

+0 = P (0)md(0)χc(0)(δ(0))σ3

(N∏

k=m+1

(d+k (0))σ3

)σ2.


Collect, now, the following facts: (1) from Lemma 4.3,

P (0) =(a+1 /a

+2 a+3 /a

+4

a+3 /a+4 a+1 /a

+2

);

(2) from the expression for md(ζ ) given in Lemma 4.1, the definition (cf. Propo-

sition 4.1) Res(χ(ζ ); ςn) =(an bncn dn

), n ∈ {1, 2, . . . , m}, and the asymptotics for

{an, bn, cn, dn}mn=1 given in Proposition 4.2, one shows that

md(0) =(

1− amςm− dm

ςm− bm

ςm− cm

ςm

− bmςm− cm

ςm1 − am

ςm− dm

ςm

)+

+O2×2(e−4t min m∈{1,2,...,N}


,

where O2×2(�) denotes a 2 × 2 matrix each of whose entries are O(�); (3) from

Lemma 4.2 and the formula for χ c12(ζ ) (= χ c

21(ζ )) given in Proposition 4.2, oneshows that

χc(0) =(

1 1√tχ c

12(0)1√tχ c

21(0) 1

)+O2×2

(cS(z0)

(z2o + 32)1/2

ln t

t

);

and (4) (δ(0))σ3(∏N

k=m+1(d+k (0))

σ3)σ2 = iδ−1(0)(∏N

k=m+1(d+k (0))

−1)σ−− iδ(0)(

∏Nk=m+1 d

+k (0))σ+. Using the results of (1)–(4), and recalling the expres-

sion for +0 given above, one arrives at

(+0)11 = a+1a+2

iδ−1(0)

(χ c

12(0)√t

(1− am

ςm− dm

ςm

)−(bm

ςm+ cm

ςm

)) N∏k=m+1

(d+k (0))−1+

+ a+3a+4

iδ−1(0)

((1− am

ςm− dm

ςm

)− χ c

12(0)√t

(bm

ςm+ cm

ςm

)) N∏k=m+1

(d+k (0))−1,

(+0)12 = a+1a+2

iδ(0)

(−(

1 − am

ςm− dm

ςm

)+ χ c

21(0)√t

(bm

ςm+ cm

ςm

)) N∏k=m+1

d+k (0)+

+ a+3a+4

iδ(0)

(− χ c

21(0)√t

(1 − am

ςm− dm

ςm

)+(bm

ςm+ cm

ςm

)) N∏k=m+1

d+k (0),

(+0)21 = a+1a+2

iδ−1(0)

((1 − am

ςm− dm

ςm

)− χ c

12(0)√t

(bm

ςm+ cm

ςm

)) N∏k=m+1

(d+k (0))−1+

+ a+3a+4

iδ−1(0)

(χ c

12(0)√t

(1− am

ςm− dm

ςm

)−(bm

ςm+ cm

ςm

)) N∏k=m+1

(d+k (0))−1,

372 A. H. VARTANIAN

(+0)22 = a+1a+2

iδ(0)

(− χ c

21(0)√t

(1 − am

ςm− dm

ςm

)+(bm

ςm+ cm

ςm

)) N∏k=m+1

d+k (0)+

+ a+3a+4

iδ(0)

(−(

1 − am

ςm− dm

ςm

)+ χ c

21(0)√t

(bm

ςm+ cm

ςm

)) N∏k=m+1

d+k (0).

Using the asymptotic expansions for {am, bm, cm, dm} (respectively, {a+i }4i=1) given

in Proposition 4.2 (respectively, Lemma 4.3), one arrives at the leading-order re-sults stated in the proposition. ✷

Remark 4.3. In Propositions 4.4 and 4.6 below, one should keep, everywhere,the upper (respectively, lower) signs for θγm = +π/2 (respectively, θγm = −π/2).

PROPOSITION 4.4. Let φ(x, t), P(φm, φk), and Q(φm) be defined by Equa-tions (67), (68), and (69), respectively. Then, for θγm = ±π/2, as t → +∞ andx →−∞ such that z0 := x/t < −2 and (x, t) ∈ �m, m ∈ {1, 2, . . . , N},a0m = −2i sin(φm)|γm|2P 2(φm, φk)Q

2(φm)e2φ(x,t)

(1− |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t)),

c0m = ∓2 sin(φm)|γm|δ−1(0)ei(φm+s+)+φ(x,t)P (φm, φk)Q(φm)

(1 − |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))

,

a1m = ∓ 16iλ2

1 sin2(φm)|γm|3√ν(λ1) P3(φm, φk)Q

3(φm) cos(s+)e3φ(x,t)

(1 − |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))2(λ21 − 2λ1 cos(φm)+ 1)2

√(λ1 − λ2) (z

20 + 32)1/4

×

×(((λ1 + λ2) cos(φm)− 2) cos

(:+(z0, t)+ π

4

)+

+ (λ1 − λ2) sin(φm) sin

(:+(z0, t)+ π

4

))∓

∓ 2λ1 sin(φm)|γm|√ν(λ1) P (φm, φk)Q(φm)eφ(x,t)

(1 − |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))(λ21 − 2λ1 cos(φm)+ 1)

√(λ1 − λ2) (z

20 + 32)1/4

×

×(

2 cos(s+) cos

(:+(z0, t)+ π

4

)− (λ1 + λ2) cos(φm + s+)×

× cos

(:+(z0, t)+ π

4

)− (λ1 − λ2) sin(φm + s+) sin

(:+(z0, t)+ π

4

)+

+ 2i sin(s+) cos

(:+(z0, t)+ π

4

)− i(λ1 + λ2) sin(φm + s+)×

× cos

(:+(z0, t)+ π

4

)+ i(λ1 − λ2) cos(φm + s+) sin

(:+(z0, t)+ π

4

)),

b1m = 2iλ1 sin(φm)|γm|2√ν(λ1) δ(0)e−i(φm+s+)+2φ(x,t)P 2(φm, φk)Q

2(φm)


√(λ1 − λ2) (z

20 + 32)1/4

×

×(

2 cos(s+) cos

(:+(z0, t)+ π

4

)− (λ1 + λ2) cos(φm + s+)×


× cos

(:+(z0, t)+ π

4

)− (λ1 − λ2) sin(φm + s+) sin

(:+(z0, t)+ π

4

)+

+ 2i sin(s+) cos

(:+(z0, t)+ π

4

)− i(λ1 + λ2) sin(φm + s+)×

× cos

(:+(z0, t)+ π

4


(:+(z0, t)+ π

4

)),

c1m = − 16λ2

1 sin2(φm)|γm|2√ν(λ1) δ−1(0)ei(φm+s+)+2φ(x,t)P 2(φm, φk)Q

2(φm) cos(s+)(1 − |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))2(λ2

1 − 2λ1 cos(φm)+ 1)2√(λ1 − λ2) (z

20 + 32)1/4

×

×(((λ1 + λ2) cos(φm)− 2) cos

(:+(z0, t)+ π

4

)+


(:+(z0, t)+ π

4

))−

− 2iλ1 sin(φm)|γm|2√ν(λ1) δ−1(0)ei(φm+s+)+2φ(x,t)P 2(φm, φk)Q

2(φm)


√(λ1 − λ2) (z

20 + 32)1/4

×

×(

2 cos(s+) cos

(:+(z0, t)+ π

4

)− (λ1 + λ2) cos(φm + s+)×

× cos

(:+(z0, t)+ π

4

)− (λ1 − λ2) sin(φm + s+) sin

(:+(z0, t)+ π

4

)−

− 2i sin(s+) cos

(:+(z0, t)+ π

4

)+ i(λ1 + λ2) sin(φm + s+)×

× cos

(:+(z0, t)+ π

4

)− i(λ1 − λ2) cos(φm+ s+) sin

(:+(z0, t)+ π

4

))+

+ 8λ21 sin2(φm)|γm|2√ν(λ1) δ

−1(0)ei(φm+s+)+2φ(x,t)P 2(φm, φk)Q2(φm)

(1 − |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))(λ21 − 2λ1 cos(φm)+ 1)2

√(λ1 − λ2) (z

20 + 32)1/4

×

×((

((λ1 + λ2) cos(φm)− 2) cos

(:+(z0, t)+ π

4

)+ (λ1 − λ2) sin(φm)×

× sin

(:+(z0, t)+ π

4

))cos(s+)− i

(((λ1 + λ2) cos(φm)− 2)×

× cos

(:+(z0, t)+ π

4

)+ (λ1 − λ2) sin(φm) sin

(:+(z0, t)+ π

4

))sin(s+)

),

d1m = ± 2λ1 sin(φm)|γm|√ν(λ1) P (φm, φk)Q(φm)eφ(x,t)


√(λ1 − λ2) (z

20 + 32)1/4

×

×(

2 cos(s+) cos

(:+(z0, t)+ π

4

)− (λ1 + λ2) cos(φm + s+)×

× cos

(:+(z0, t)+ π

4

)− (λ1 − λ2) sin(φm + s+) sin

(:+(z0, t)+ π

4

)+

374 A. H. VARTANIAN

+ 2i sin(s+) cos

(:+(z0, t)+ π

4

)− i(λ1 + λ2) sin(φm + s+)×

× cos

(:+(z0, t)+ π

4


(:+(z0, t)+ π

4

)),

where s+ is given in Theorem 2.2.1, Equation (11).Proof. Recalling the definitions of {a0

m, a1m, b

1m, c

0m, c

1m, d

1m} given in Proposi-

tion 4.2, substituting into them the expressions for g∗m and χ c12(ζ ) given in Proposi-

tions 4.1 and 4.2, respectively, using standard trigonometric identities, and definingφ(x, t), P(φm, φk), and Q(φm) as in Equations (67), (68), and (69), respectively,one obtains, after tedious, but otherwise straightforward calculations, the resultstated in the proposition.

PROPOSITION 4.5. As t → +∞ and x → −∞ such that z0 := x/t < −2 and(x, t) ∈ �m, m ∈ {1, 2, . . . , N},

u(x, t) = i

((+0)12 + a+3 + bm + cm +

√ν(λ1) e

i<+(0)2 λ

2iν(λ1)1√

t (λ1 − λ2) (z20 + 32)1/4

×

× (λ1e−i(:+(z0,t )+ π4 ) + λ2ei(:+(z0,t )+ π

4 ))

)+O

(cS(z0)

(z20 + 32)1/2

ln t

t

), (70)∫ x

+∞(|u(x′, t)|2 − 1) dx′

= −i

((+0)11 + a+1 − a+2 + am + dm + 2i

N∑k=m+1

sin(φk)+

+ i

(∫ 0

−∞+∫ λ1

λ2

)ln(1 − |r(µ)|2) dµ

2π

)+ O

(cS(z0)

(z20 + 32)1/2

ln t

t

), (71)∫ x

−∞(|u(x′, t)|2 − 1) dx′

=∫ x

+∞(|u(x′, t)|2 − 1) dx′ − 2

N∑n=1

sin(φn)−∫ +∞

−∞ln(1− |r(µ)|2) dµ

2π. (72)

Proof. Recall Equations (63), (64), and (65) for u(x, t),∫ x

+∞(|u(x′, t)|2−1) dx′,and χ(ζ ), respectively. Using the result for P (ζ ) (respectively, χc

ij (ζ ), i, j ∈{1, 2}) stated in Lemma 4.3 (respectively, Lemma 4.2), noting that

(δ(ζ ))±1 =ζ→∞ 1 ± i

((∫ 0

−∞+∫ λ1

λ2

)ln(1 − |r(µ)|2)dµ

2π

)ζ−1 +O(ζ−2)

andN∏

k=m+1

(d+k (ζ ))±1 =

ζ→∞ 1 ±(

N∑k=m+1

(ςk − ςk)

)ζ−1 +O(ζ−2),


and using the asymptotic estimates for {an, bn, cn, dn}m−1n=1 given in Proposition 4.2,

one forms the large-ζ asymptotics for χ (ζ ) given in Equation (65) to show that

(ζ(χ(ζ )(δ(ζ ))σ3

N∏k=m+1

(d+k (ζ ))σ3 − I))11

=ζ→∞

ζ∈C\(σd∪σc)(+o)11 + a+1 − a+2 + am + dm +

N∑k=m+1

(ςk − ςk)+

+ i

(∫ 0

−∞+∫ λ1

λ2

)ln(1 − |r(µ)|2)dµ

2π+

+O

(cS(z0)

(z20 + 32)1/2

ln t

t

)+O(e−Qt ),

(ζ(χ(ζ )(δ(ζ ))σ3

N∏k=m+1

(d+k (ζ ))σ3 − I))12

=ζ→∞

ζ∈C\(σd∪σc)(+0)12 + a+3 + bm + cm +

√ν(λ1) e

i<+(0)2 λ

2iν(λ1)1√

t (λ1 − λ2) (z20 + 32)1/4

×

× (λ1e−i(:+(z0,t )+ π4 ) + λ2ei(:+(z0,t )+ π

4 ))++O

(cS(z0)

(z20 + 32)1/2

ln t

t

)+O(e−Qt ),

where Q := 4 min m∈{1,2,...,N}n�=m∈{1,2,...,N}

{sin(φn)|cos(φn) − cos(φm)|} (> 0). Neglecting ex-

ponentially small terms (cf. Remark 4.1), from the expressions for u(x, t) and∫ x

+∞(|u(x′, t)|2 − 1) dx′ given, respectively, in Equations (63) and (64), and thetrace identity (cf. Equation (4))∫ +∞

−∞(|u(x′, t)|2 − 1) dx′ =

(∫ x

−∞+∫ +∞

x

)(|u(x′, t)|2 − 1) dx′

= −2N∑n=1

sin(φn)−∫ +∞

−∞ln(1− |r(µ)|2) dµ

2π,

one obtains the results stated in the proposition. ✷PROPOSITION 4.6. As t → +∞ and x → −∞ such that z0 := x/t < −2 and(x, t) ∈ �m, m ∈ {1, 2, . . . , N}, for θγm = ±π/2,

(+0)11 = i

(± 2 sin(φm)|γm|P(φm, φk)Q(φm)eφ(x,t)

(1 − |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))

+

+√ν(λ1)√

t (λ1 − λ2) (z20 + 32)1/4

(−2 cos

(:+(z0, t)+ π

4

)cos(s+)+

376 A. H. VARTANIAN

+ 4 sin(φm)|γm|2P 2(φm, φk)Q2(φm) sin(s+ − φm) cos(:+(z0, t)+ π

4 )e2φ(x,t)

(1 − |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))+

+ 8λ21 sin2(φm)|γm|2P 2(φm, φk)Q

2(φm)(1 + |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))e2φ(x,t)


×

×(((λ1 + λ2) cos(φm)− 2) cos

(:+(z0, t)+ π

4

)+


(:+(z0, t)+ π

4

))cos(s+)+

+ 4λ1 sin(φm) cos(φm)|γm|2P 2(φm, φk)Q2(φm)e2φ(x,t)


×

×(

2 sin(s+ − φm) cos

(:+(z0, t)+ π

4

)− (λ1 + λ2) sin(s+)×

× cos

(:+(z0, t)+ π

4

)+ (λ1 − λ2) cos(s+) sin

(:+(z0, t)+ π

4

))))+

+O

(cS(z0)

(z20 + 32)1/2

ln t

t

),

(+0)12 = −ie−i(θ+(1)+s+)++ 2 sin(φm)|γm|2P 2(φm, φk)Q

2(φm)e−i(θ+(1)+φm+s+)+2φ(x,t)

(1− |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))+

+ 1√t

(2i Im(a1 − a2) sin(φm)|γm|2P 2(φm, φk)Q

2(φm)e−i(θ+ (1)+φm+s+)+2φ(x,t)

(1 − |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))±

± 4i sin(φm)|γm|P (φm, φk)Q(φm)√ν(λ1) e−i(θ+ (1)+2s+ )+φ(x,t) cos(:+(z0, t)+ π

4 )

(1 − |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))√(λ1 − λ2) (z

20 + 32)1/4

∓

∓ 16iλ21 sin2(φm)|γm|3P 3(φm, φk)Q

3(φm)√ν(λ1) e−i(θ+(1)+s+)+3φ(x,t) cos(s+)


√(λ1 − λ2) (z

20 + 32)1/4

×

×(((λ1 + λ2) cos(φm)− 2) sin(φm) cos

(:+(z0, t)+ π

4

)+

+ (λ1 − λ2) sin2(φm) sin

(:+(z0, t)+ π

4

)+

+ i

(((λ1 + λ2) cos(φm)− 2) cos(φm) cos

(:+(z0, t)+ π

4

)+

+ (λ1 − λ2) sin(φm) cos(φm) sin

(:+(z0, t)+ π

4

))+

+ Im(a1 − a2)e−i(θ+(1)+s+)−

− 4λ1 sin(φm)|γm|P(φm, φk)Q(φm)√ν(λ1) e−i(θ+(1)+s+ )+φ(x,t)


√(λ1 − λ2) (z

20 + 32)1/4

×

×(∓ 2 sin(s+ − φm) cos

(:+(z0, t)+ π

4

)± (λ1 + λ2) sin(s+)×


× cos

(:+(z0, t)+ π

4

)∓ (λ1 − λ2) cos(s+) sin

(:+(z0, t)+ π

4

)))+

+O

(cS(z0)

(z20 + 32)1/2

ln t

t

),

Im(a1 − a2) = ±√ν(λ1) sin(s+) cos(:+(z0, t)+ π

4 )(1 − |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))√

(λ1 − λ2) (z20 + 32)1/4 sin(φm)|γm|P (φm, φk)Q(φm)eφ(x,t)

±

± 4λ21

√ν(λ1) sin(φm)|γm|P(φm, φk)Q(φm) sin(s+)eφ(x,t)

(λ21 − 2λ1 cos(φm)+ 1)2

√(λ1 − λ2) (z

20 + 32)1/4

×

×(((λ1 + λ2) cos(φm)− 2) cos

(:+(z0, t)+ π

4

)+


(:+(z0, t)+ π

4

))±

± 2λ1√ν(λ1) cos(φm)|γm|P(φm, φk)Q(φm)eφ(x,t)

(λ21 − 2λ1 cos(φm)+ 1)

√(λ1 − λ2) (z

20 + 32)1/4

×

×(

2 cos(s+ − φm) cos

(:+(z0, t)+ π

4

)−

− (λ1 + λ2) cos(s+) cos

(:+(z0, t)+ π

4

)−

− (λ1 − λ2) sin(s+) sin

(:+(z0, t)+ π

4

))±

± 2√ν(λ1)|γm|P (φm, φk)Q(φm) cos(s+ − φm) cos(:+(z0, t)+ π

4 )eφ(x,t)

√(λ1 − λ2) (z

20 + 32)1/4

+

+ O

(cS(z0)

(z20 + 32)1/2

ln t

t

),

Re(a1 − a2) = Re(a3) = Im(a3) = 0,

where θ+(·) is given in Theorem 2.2.1, Equation (8).Proof. Recall from Lemma 4.1 that: (1) +0 = P (0)md(0)χc(0)(δ(0))σ3 ×

(∏N

k=m+1(d+k (0))

σ3)σ2; (2) tr(+0) = 0; (3) det(+0) = −1; and (4) +0+0 = I.Taking the determinant of both sides of the above expression for +0 and using thefact that det(+0) = −1, it follows that, modulo terms that are O(

cS(z0)

(z20+32)1/2

ln tt), and

always ignoring exponentially small terms, det(P (0)) = (det(md(0)))−1. Beforeproceeding further, this will be verified; in particular, since md(ζ ) ∈ SL(2,C), itmust be the case that, modulo terms that are

O

(cS(z0)

(z20 + 32)1/2

ln t

t

), det(md(0)) = 1.

378 A. H. VARTANIAN

From Lemma 4.3, keeping only leading-order terms, one shows that

det(P (0)) = 1 + 2Re(a1 − a2)√t

+O

(cS(z0)

(z20 + 32)1/2

ln t

t

),

and, from the proof of Proposition 4.1, the estimates of Proposition 4.2, and notingthat (

1− a0m

ςm

)(1 − a0

m

ςm

)− c0

m

ςm

c0m

ςm= 1

and ∫ +∞

−∞(1− µ2) ln(1 − |r(µ)|2)(µ2 − 2µ cos(φm)+ 1)

dµ

µ= 0

(which is proven using the symmetry reduction r(ζ−1) = −r(ζ )), one shows that

(det(md(0)))−1

= 1 + 2√tRe

((1 − a0

m

ςm

)(a1m

ςm+ d1

m

ςm

)+ c0

m

ςm

(b1m

ςm+ c1

m

ςm

))+

+O

(cS(z0)

(z20 + 32)1/2

ln t

t

);

thus, from the – yet to be verified – identity det(P (0)) = (det(md(0)))−1, and theabove, it follows that

Re(a1 − a2) = Re

((1− a0

m

ςm

)(a1m

ςm+ d1

m

ςm

)+ c0

m

ςm

(b1m

ςm+ c1

m

ςm

)).

If the formulae presented thus far are correct, then one must be able to show fromthem that the right-hand side of the latter relation equals zero. From Proposition 4.2and repeated application of standard trigonometric identities, one shows, after avery lengthy and tedious algebraic calculation, that (θγm = ±π/2)

Re

(c0m

ςm

(b1m

ςm+ c1

m

ςm

))= ±16λ2

1 sin3(φm)|γm|3P 3(φm, φk)Q3(φm)

√ν(λ1) (1 + |γm|2P 2(φm, φk)Q

2(φm)e2φ(x,t))e3φ(x,t)

(λ21 − 2λ1 cos(φm)+ 1)2

√(λ1 − λ2) (z

20 + 32)1/4(1 − |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))3

×

×(((λ1 + λ2) cos(φm)− 2) cos

(:+(z0, t)+ π

4

)+


(:+(z0, t)+ π

4

))cos(s+)±

± 8λ1 sin2(φm) cos(φm)|γm|3P 3(φm, φk)Q3(φm)

√ν(λ1) e3φ(x,t)

(1 − |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))2(λ21 − 2λ1 cos(φm)+ 1)

√(λ1 − λ2) (z

20 + 32)1/4

×


×(


(:+(z0, t)+ π

4

)− (λ1 + λ2) sin(s+)×

× cos

(:+(z0, t)+ π

4

)+ (λ1 − λ2) cos(s+) sin

(:+(z0, t)+ π

4

)),

and

Re

((1− a0

m

ςm

)(a1m

ςm+ d1

m

ςm

))= ∓16λ2


√ν(λ1) (1 + |γm|2P 2(φm, φk)Q

2(φm)e2φ(x,t))e3φ(x,t)

(λ21 − 2λ1 cos(φm)+ 1)2

√(λ1 − λ2) (z

20 + 32)1/4(1 − |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))3

×

×(((λ1 + λ2) cos(φm)− 2) cos

(:+(z0, t)+ π

4

)+


(:+(z0, t)+ π

4

))cos(s+)∓

∓ 8λ1 sin2(φm) cos(φm)|γm|3P 3(φm, φk)Q3(φm)



√(λ1 − λ2) (z

20 + 32)1/4

×

×(


(:+(z0, t)+ π

4

)− (λ1 + λ2) sin(s+)×

× cos

(:+(z0, t)+ π

4

)+ (λ1 − λ2) cos(s+) sin

(:+(z0, t)+ π

4

));

thus, adding,

Re

((1 − a0

m

ςm

)(a1m

ςm+ d1

m

ςm

)+ c0

m

ςm

(b1m

ςm+ c1

m

ςm

))= 0,

whence Re(a1 − a2) = 0. Recalling the expression for (+0)11 given in Propo-sition 4.3, the estimates and expansions of Proposition 4.2, and the fact – justestablished – that Re(a1 − a2) = 0, one shows that

(+0)11 = 1√t(+0)

α11 + i

((+0)

β

11 +1√t(+0)

γ

11

)+ O

(cS(z0)

(z20 + 32)1/2

ln t

t

),

where

(+0)α11

:= ∓2 Im(a1 − a2) sin(φm)|γm|P(φm, φk)Q(φm)eφ(x,t)

(1 − |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))+

+ 2√ν(λ1) cos(:+(z0, t)+ π

4 ) sin(s+)√(λ1 − λ2) (z

20 + 32)1/4

−− Re(a3) sin(θ+(1)+ s+)− Im(a3) cos(θ+(1)+ s+)+

380 A. H. VARTANIAN

+ 4 sin(φm)|γm|2P 2(φm, φk)Q2(φm)

√ν(λ1) cos(:+(z0, t)+ π

4 ) cos(s+ − φm)e2φ(x,t)


20 + 32)1/4

+

+ 2 Re(a3) sin(φm)|γm|2P 2(φm, φk)Q2(φm) cos(s+ + φm + θ+(1))e2φ(x,t)

(1 − |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))−

− 2 Im(a3) sin(φm)|γm|2P 2(φm, φk)Q2(φm) sin(s+ + φm + θ+(1))e2φ(x,t)

(1 − |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))+


2(φm)√ν(λ1) sin(s+)e2φ(x,t)

(1 − |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))(λ21 − 2λ1 cos(φm)+ 1)2

√(λ1 − λ2) (z

20 + 32)1/4

×

×(((λ1 + λ2) cos(φm)− 2) cos

(:+(z0, t)+ π

4

)+


(:+(z0, t)+ π

4

))+

+ 4λ1 sin(φm) cos(φm)|γm|2P 2(φm, φk)Q2(φm)



√(λ1 − λ2) (z

20 + 32)1/4

×

×(

2 cos(s+ − φm) cos

(:+(z0, t)+ π

4

)− (λ1 + λ2) cos(s+)×

× cos

(:+(z0, t)+ π

4

)− (λ1 − λ2) sin(s+) sin

(:+(z0, t)+ π

4

)),

(+0)β

11 := ± 2 sin(φm)|γm|P(φm, φk)Q(φm)eφ(x,t)

(1 − |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t)),

(+0)γ

11

:= −2√ν(λ1) cos(:+(z0, t)+ π

4 ) cos(s+)√(λ1 − λ2) (z

20 + 32)1/4

++ Re(a3) cos(θ+(1)+ s+)− Im(a3) sin(θ+(1)+ s+)++ 4 sin(φm)|γm|2P 2(φm, φk)Q

2(φm)√ν(λ1) cos(:+(z0, t)+ π

4 ) sin(s+ − φm)e2φ(x,t)


20 + 32)1/4

+

+ 2 Re(a3) sin(φm)|γm|2P 2(φm, φk)Q2(φm) sin(s+ + φm + θ+(1))e2φ(x,t)

(1 − |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))+

+ 2 Im(a3) sin(φm)|γm|2P 2(φm, φk)Q2(φm) cos(s+ + φm + θ+(1))e2φ(x,t)

(1 − |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))+


2(φm)√ν(λ1) cos(s+)

(λ21 − 2λ1 cos(φm)+ 1)2

√(λ1 − λ2) (z

20 + 32)1/4

×

× (1 + |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))e2φ(x,t)

(1− |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))2×

×(((λ1 + λ2) cos(φm)− 2) cos

(:+(z0, t)+ π

4

)+


(:+(z0, t)+ π

4

))+


+ 4λ1 sin(φm) cos(φm)|γm|2P 2(φm, φk)Q2(φm)



√(λ1 − λ2) (z

20 + 32)1/4

×

×(


(:+(z0, t)+ π

4

)− (λ1 + λ2) sin(s+)×

× cos

(:+(z0, t)+ π

4

)+ (λ1 − λ2) cos(s+) sin

(:+(z0, t)+ π

4

)),

and θ+(·) is specified in the proposition. Recalling that tr(+0) = 0, it follows thatRe((+0)11) = 0; thus, (+0)

α11 = 0, which gives a relation for Im(a1 − a2), but,

since Re(a3) and Im(a3) are as yet undetermined, this is not enough. Towards this

end, one uses the condition det(+0) = (+0)11(+0)11 − (+0)12(+0)12 = −1 (Note:if the conditions tr(+0) = 0 and det(+0) = −1 are satisfied, then it follows that+0+0 = I is also satisfied, so it is enough to use the condition det(+0) = −1).From the formula for (+0)11 given above, and the expression for (+0)12 given inProposition 4.3, one shows that

(+0)12(+0)12 =(

1− a0m

ςm

)(1 − a0

m

ςm

)+ 2√

t

(Re

((a1 − a2)

(1 − a0

m

ςm

)×

×(

1− a0m

ςm

))− Re

((1 − a0

m

ςm

)(a1m

ςm+ d1

m

ςm

))+

+ Re

((1− a0

m

ςm

)2c0

m

√ν(λ1) δ(0) cos(:+(z0, t)+ π

4 )

ςm√(λ1 − λ2) (z

20 + 32)1/4

)−

− Re

((1− a0

m

ςm

)c0ma3

ςm

))+O

(cS(z0)

(z20 + 32)1/2

ln t

t

).

Using the estimates given in Proposition 4.2, and recalling that Re(a1 − a2) = 0,one gets that

(+0)12(+0)12

= 1 + 4 sin2(φm)|γm|2P 2(φm, φk)Q2(φm)e2φ(x,t)

(1 − |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))

+

+ 4 sin2(φm)|γm|4P 4(φm, φk)Q4(φm)e4φ(x,t)

(1− |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))2+

+ 2√t

(∓ 4 sin(φm)|γm|P (φm, φk)Q(φm)

√ν(λ1) cos(s+) cos(:+(z0, t)+ π

4 )eφ(x,t)


20 + 32)1/4

±

± 8 sin2(φm)|γm|3P 3(φm, φk)Q3(φm)

√ν(λ1) sin(s+ − φm) cos(:+(z0, t)+ π

4 )e3φ(x,t)

(1 − |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))2√(λ1 − λ2) (z

20 + 32)1/4

±

± 2 sin(φm)|γm|P(φm, φk)Q(φm)eφ(x,t)

(1 − |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))(Re(a3) cos(s+ + θ+(1))−

382 A. H. VARTANIAN

− Im(a3) sin(s+ + θ+(1)))± 4 sin2(φm)|γm|3P 3(φm, φk)Q3(φm)e3φ(x,t)

(1 − |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))2×

× (Re(a3) sin(s+ + θ+(1)+ φm)+ Im(a3) cos(s+ + θ+(1)+ φm))±± 16λ2


√ν(λ1) cos(s+)

(λ21 − 2λ1 cos(φm)+ 1)2

√(λ1 − λ2) (z

20 + 32)1/4

×


(1− |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))3

×

×(((λ1 + λ2) cos(φm)− 2) cos

(:+(z0, t)+ π

4

)+


(:+(z0, t)+ π

4

))±




√(λ1 − λ2) (z

20 + 32)1/4

×

×(


(:+(z0, t)+ π

4

)−

− (λ1 + λ2) sin(s+) cos

(:+(z0, t)+ π

4

)+

+ (λ1 − λ2) cos(s+) sin

(:+(z0, t)+ π

4

)))+O

(cS(z0)

(z20 + 32)1/2

ln t

t

).

From the expression for (+0)11 given above, and using the fact that (+0)α11 = 0,

one shows that

(+0)11(+0)11

= 4 sin2(φm)|γm|2P 2(φm, φk)Q2(φm)e2φ(x,t)

(1− |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))2

+

+ 2√t

(∓ 4 sin(φm)|γm|P (φm, φk)Q(φm)

√ν(λ1) cos(s+) cos(:+(z0, t)+ π

4 )eφ(x,t)


20 + 32)1/4

±

± 8 sin2(φm)|γm|3P 3(φm, φk)Q3(φm)

√ν(λ1) sin(s+ − φm) cos(:+(z0, t)+ π

4 )e3φ(x,t)

(1 − |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))2√(λ1 − λ2) (z

20 + 32)1/4

±

± 2 Re(a3) sin(φm)|γm|P(φm, φk)Q(φm)eφ(x,t)

(1 − |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))

(cos(s+ + θ+(1))+

+ 2 sin(φm)|γm|2P 2(φm, φk)Q2(φm) sin(s+ + φm + θ+(1))e2φ(x,t)

(1− |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))

)±

± 2 Im(a3) sin(φm)|γm|P(φm, φk)Q(φm)eφ(x,t)

(1 − |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))

(− sin(s+ + θ+(1))+


+ 2 sin(φm)|γm|2P 2(φm, φk)Q2(φm) cos(s+ + φm + θ+(1))e2φ(x,t)

(1 − |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))

)±

± 16λ21 sin3(φm)|γm|3P 3(φm, φk)Q

3(φm)√ν(λ1) cos(s+)

(λ21 − 2λ1 cos(φm)+ 1)2

√(λ1 − λ2) (z

20 + 32)1/4

×


(1− |γm|2P 2(φm, φk)Q2(φm)e2φ(x,t))3×

×(((λ1 + λ2) cos(φm)− 2) cos

(:+(z0, t)+ π

4

)+


(:+(z0, t)+ π

4

))±




√(λ1 − λ2) (z

20 + 32)1/4

×

×(


(:+(z0, t)+ π

4

)−

− (λ1 + λ2) sin(s+) cos

(:+(z0, t)+ π

4

)+

+ (λ1 − λ2) cos(s+) sin

(:+(z0, t)+ π

4

)))+O

(cS(z0)

(z20 + 32)1/2

ln t

t

).

Now, taking note of the relation

(+0)β

11(+0)β

11 −(

1− a0m

ςm

)(1 − a0

m

ςm

)= −1,

one substitutes the above-derived formulae for |(+0)11|2 and |(+0)12|2 into|(+0)11|2 −|(+0)12|2 = −1, and, modulo terms that are O(

cS (z0)

(z20+32)1/2

ln tt), gets exact

cancellation at O(1) and O(t−1/2); thus, one concludes that Re(a3) = Im(a3) = 0.Recalling that (+0)

α11 = 0, and using the fact that Re(a3) = Im(a3) = 0, from

the expression for (+0)11 given above, one obtains, after some straightforwardalgebra, the expressions for Im(a1− a2) and (+0)11 stated in the Proposition. FromProposition 4.2, and the fact that Re(a3) = Im(a3) = Re(a1− a2) = 0, one obtains,upon recalling the expression for (+0)12 given in Proposition 4.3, the formula for(+0)12 given in the proposition. ✷LEMMA 4.4. As t → +∞ and x → −∞ such that z0 := x/t < −2 and(x, t) ∈ �m, m ∈ {1, 2, . . . , N}, u(x, t), the solution of the Cauchy problemfor the Df NLSE, and

∫ x

±∞(|u(x′, t)|2 − 1) dx′ have the leading-order asymptoticexpansions (for the upper sign) stated in Theorem 2.2.1, Equations (7)–(20).

Proof. The asymptotic expansions for u(x, t) and∫ x

±∞(|u(x′, t)|2−1) dx′ followfrom Proposition 4.2, Proposition 4.4, Equations (70)–(72), and Proposition 4.6after tedious, but otherwise straightforward algebraic calculations. ✷

384 A. H. VARTANIAN

Appendix A. Asymptotic Analysis as t → −∞In this appendix, a silhouette of the asymptotic analysis for u(x, t) and∫ x

±∞(|u(x′, t)|2 − 1) dx′ as t →−∞ and x → +∞ such that z0 := x/t < −2 and(x, t) ∈ �m, m ∈ {1, 2, . . . , N}, is presented. Since the calculations are analogousto those of Sections 3 and 4, only final results/statements, with in one instance asketch of a proof, are given: one mimics the scheme of the calculation in Sections 3and 4 to arrive at the corresponding asymptotic results.

The analogue of Lemma 3.1 is

LEMMA A.1.1. For r(ζ ) ∈ S1C(R), let m(ζ): C \ (σd ∪ σc) → M2(C) be the

solution of the RHP formulated in Lemma 2.1.2. Set m(ζ ) := m(ζ)(δ(ζ ))−σ3 ,where

δ(ζ ) = exp

((∫ λ2

0+∫ +∞

λ1

)ln(1 − |r(µ)|2)

(µ− ζ )

dµ

2π i

),

with λ1 and λ2 given in Theorem 2.2.1, Equation (10), δ(ζ )δ(ζ ) = 1, δ(ζ )δ(ζ−1) =δ(0), and ‖(δ(·))±1‖L∞(C) := supζ∈C |(δ(ζ ))±1| <∞. Then m(ζ ): C\(σd∪σc)→M2(C) solves the following RHP:

(i) m(ζ ) is piecewise (sectionally) meromorphic ∀ζ ∈ C \ σc;(ii) m±(ζ ) := lim ζ ′→ζ

±Im(ζ ′)>0m(ζ ′) satisfy the jump condition

m+(ζ ) = m−(ζ ) exp(−ik(ζ )(x + 2λ(ζ )t) ad(σ3))G(ζ ), ζ ∈ R,

where

G(ζ ) =((1− r(ζ )r(ζ ))δ−(ζ )(δ+(ζ ))−1 −r(ζ ) δ−(ζ )δ+(ζ )

r(ζ )(δ−(ζ )δ+(ζ ))−1 (δ−(ζ ))−1δ+(ζ )

);

(iii) m(ζ ) has simple poles in σd = ⋃Nn=1({ςn} ∪ {ςn}) with

Res(m(ζ ); ςn) = limζ→ςn

m(ζ )gn(δ(ςn))−2σ−, n ∈ {1, 2, . . . , N},

Res(m(ζ ); ςn) = σ1Res(m(ζ ); ςn) σ1, n ∈ {1, 2, . . . , N},where gn is defined in Lemma 3.1(iii);

(iv) det(m(ζ ))|ζ=±1 = 0;(v) m(ζ ) =ζ→0 ζ

−1(δ(0))σ3σ2 +O(1);(vi) m(ζ ) = ζ→∞


(vii) m(ζ ) = σ1m(ζ ) σ1 and m(ζ−1) = ζ m(ζ )(δ(0))σ3σ2.

Let


ζ∈C\(σd∪σc)(ζ(m(ζ )(δ(ζ ))σ3 − I))12, (73)


and ∫ x

+∞(|u(x′, t)|2 − 1) dx′ := −i lim

ζ→∞ζ∈C\(σd∪σc)

(ζ(m(ζ )(δ(ζ ))σ3 − I))11. (74)


The analogue of Definition 3.1 is

DEFINITION A.1.1. For m ∈ {1, 2, . . . , N} and {ςn}m−1n=1 ⊂ C+ (respectively,

{ςn}m−1n=1 ⊂ C−), define the clockwise (respectively, counter-clockwise) oriented

circles Kn := {ζ ; |ζ − ςn| = εKn } (respectively, Ln := {ζ ; |ζ − ςn| = εL

n }), withεKn (respectively, εL

n ) chosen sufficiently small such that Kn ∩ Kn′ = Ln ∩ Ln′ =Kn ∩ Ln = Kn ∩ σc = Ln ∩ σc = ∅ ∀n �= n′ ∈ {1, 2, . . . , m− 1}.


LEMMA A.1.2. For r(ζ ) ∈ S1C(R), let m(ζ ): C \ (σd ∪ σc) → M2(C) be the

solution of the RHP formulated in Lemma A.1.1. Set

m6(ζ ) :=

m(ζ ), ζ ∈ C \ (σc ∪ (

⋃m−1n=1 (Kn ∪ int(Kn) ∪ Ln ∪ int(Ln)))),

m(ζ )(I − gn(δ(ςn))

−2

(ζ−ςn) σ−), ζ ∈ int(Kn), n ∈ {1, 2, . . . , m− 1},

m(ζ )(I + gn(δ(ςn))−2

(ζ−ςn) σ+), ζ ∈ int(Ln), n ∈ {1, 2, . . . , m− 1}.

Then m6(ζ ): C\ ((σd \⋃m−1n=1 ({ςn}∪ {ςn}))∪ (σc∪ (

⋃m−1n=1 (Kn∪ Ln))))→ M2(C)

solves the following RHP:

(i) m6(ζ ) is piecewise (sectionally) meromorphic ∀ζ ∈ C \ (σc ∪ (⋃m−1

n=1 (Kn ∪Ln)));

(ii) m6±(ζ ) := lim ζ ′→ζ

ζ ′∈± side of σc∪(⋃m−1n=1 (Kn∪Ln))

m6(ζ ′) satisfy the jump condition

m6+(ζ ) = m

6−(ζ )υ6(ζ ), ζ ∈ σc ∪

(m−1⋃n=1

(Kn ∪ Ln)

),

where

υ6(ζ ) =

exp(−ik(ζ )(x + 2λ(ζ )t) ad(σ3))G(ζ ), ζ ∈ R,

I + gn(δ(ςn))−2

(ζ−ςn) σ−, ζ ∈ Kn, n ∈ {1, 2, . . . , m− 1},

I + gn(δ(ςn))−2

(ζ−ςn) σ+, ζ ∈ Ln, n ∈ {1, 2, . . . , m− 1},with G(ζ ) given in Lemma A.1.1(ii);

386 A. H. VARTANIAN

(iii) m6(ζ ) has simple poles in σd \⋃m−1n=1 ({ςn} ∪ {ςn}) with


m6(ζ )gn(δ(ςn))−2σ−,

n ∈ {m,m+ 1, . . . , N},Res(m6(ζ ); ςn) = σ1Res(m6(ζ ); ςn) σ1, n ∈ {m,m+ 1, . . . , N};

(iv) det(m6(ζ ))|ζ=±1 = 0;(v) m6(ζ ) =ζ→0 ζ

−1(δ(0))σ3σ2 +O(1);(vi) as ζ →∞, ζ ∈ C \ ((σd \⋃m−1

n=1 ({ςn} ∪ {ςn}))∪ (σc ∪ (⋃m−1

n=1 (Kn ∪ Ln)))),m6(ζ ) = I +O(ζ−1);

(vii) m6(ζ ) = σ1m6(ζ ) σ1 and m6(ζ−1) = ζ m6(ζ )(δ(0))σ3σ2.

For ζ ∈ C \ ((σd \⋃m−1n=1 ({ςn} ∪ {ςn})) ∪ (σc ∪ (

⋃m−1n=1 (Kn ∪ Ln)))), let

u(x, t) := i limζ→∞(ζ(m

6(ζ )(δ(ζ ))σ3 − I))12, (75)

and ∫ x

+∞(|u(x′, t)|2 − 1) dx′ := −i lim

ζ→∞(ζ(m(ζ )(δ(ζ ))σ3 − I))11. (76)



LEMMA A.1.3. For m ∈ {1, 2, . . . , N}, let σ ′′d := σd \⋃m−1

n=1 ({ςn} ∪ {ςn}), σ ′′c :=

σc ∪ (⋃m−1

n=1 (Kn ∪ Ln)), where Kn and Ln are given in Definition A.1.1, andσ ′′

OD := σ ′′d ∪ σ ′′

c (σ ′′d ∩ σ ′′

c = ∅). Set

m8(ζ ) :=

m6(ζ )∏m−1

k=1 (d+k (ζ ))

−σ3,

ζ ∈ C \ (σ ′′c ∪ (

⋃m−1n=1 (int(Kn) ∪ int(Ln)))),

m6(ζ )(JKn(ζ ))−1

∏m−1k=1 (d

−k (ζ ))

−σ3,

ζ ∈ int(Kn), n ∈ {1, 2, . . . , m− 1},m6(ζ )(JLn

(ζ ))−1∏m−1k=1 (d

−k (ζ ))

−σ3,

ζ ∈ int(Ln), n ∈ {1, 2, . . . , m− 1},where d±n (ζ ) are given in Lemma 3.3, JKn

(ζ ) (∈ SL(2,C)) and JLn(ζ ) (∈ SL(2,

C)), respectively, are holomorphic in⋃m−1

k=1 int(Kk) and⋃m−1

l=1 int(Ll), with

JKn(ζ ) =

∏m−1

k=1k �=n

d+k(ζ)

d−k(ζ)− CK

n gn(δ(ςn))−2

(ζ−ςn)2∏m−1

k=1k �=n

(d+k(ζ))−1

d−k(ζ)

(ζ−ςn)CKn

(ζ−ςn)2

∏m−1k=1k �=n

(d+k (ζ ))−1

d−k (ζ )

−gn(δ(ςn))−2∏m−1k=1k �=n

d−k (ζ )d+k (ζ )

(ζ − ςn)∏m−1

k=1k �=n

d−k (ζ )d+k (ζ )

,


JLn(ζ ) =

(ζ − ςn)∏m−1

k=1k �=n

d+k (ζ )d−k (ζ )

gn(δ(ςn))−2∏m−1

k=1k �=n

d+k (ζ )d−k (ζ )

− CLn

(ζ−ςn)2

∏m−1k=1k �=n

d−k (ζ )(d+k (ζ ))−1

∏m−1k=1k �=n

d−k(ζ)

d+k(ζ)− CL

n gn(δ(ςn))−2

(ζ−ςn)2∏m−1

k=1k �=n

d−k(ζ)

(d+k(ζ))−1

(ζ−ςn)

,

and

CKn = CL

n = −4 sin2(φn)(gn)−1(δ(ςn))

2 e−2i

∑m−1j=1j �=n

φj×

×m−1∏k=1k �=n

(sin( 1

2(φn + φk))

sin( 12(φn − φk))

)2

.

Then m8(ζ ): C \ σ ′′OD → M2(C) solves the following (augmented) RHP:

(i) m8(ζ ) is piecewise (sectionally) meromorphic ∀ζ ∈ C \ σ ′′c ;

(ii) m8±(ζ ) := lim ζ ′→ζ

ζ ′∈± side of σ ′′OD

m8(ζ ′) satisfy the following jump conditions,

m8+(ζ ) = m

8−(ζ ) exp(−ik(ζ )(x + 2λ(ζ )t) ad(σ3))G

8(ζ ), ζ ∈ R,

where

G8(ζ )

=(

(1 − r(ζ )r(ζ ))δ−(ζ )(δ+(ζ ))−1 −r(ζ ) δ−(ζ )δ+(ζ )∏m−1k=1 (d+k (ζ ))2

r(ζ )(δ−(ζ )δ+(ζ ))−1∏m−1k=1 (d+k (ζ ))−2 (δ−(ζ ))−1δ+(ζ )

),

and

m8+(ζ ) =

{m

8−(ζ )

(I + CK

n

(ζ−ςn)σ+), ζ ∈ Kn, n ∈ {1, 2, . . . , m− 1},

m8−(ζ )

(I + CL

n

(ζ−ςn)σ−), ζ ∈ Ln, n ∈ {1, 2, . . . , m− 1};

(iii) m8(ζ ) has simple poles in σ ′′d with


m8(ζ )gn(δ(ςn))−2

(m−1∏k=1

(d+k (ςn))−2

)σ−,

n ∈ {m,m+ 1, . . . , N},Res(m8(ζ ); ςn) = σ1Res(m8(ζ ); ςn) σ1, n ∈ {m,m+ 1, . . . , N};

(iv) det(m8(ζ ))|ζ=±1 = 0;(v) m8(ζ ) =ζ→0 ζ

−1(δ(0))σ3(∏m−1

k=1 (d+k (0))

σ3)σ2 +O(1);(vi) m8(ζ ) = ζ→∞

ζ∈C\σ ′′OD

I +O(ζ−1);

(vii) m8(ζ ) = σ1m8(ζ ) σ1 and m8(ζ−1) = ζ m8(ζ )(δ(0))σ3(∏m−1

k=1 (d+k (0))

σ3)σ2.

388 A. H. VARTANIAN

Let


ζ∈C\σ ′′OD

(ζ

(m8(ζ )(δ(ζ ))σ3

m−1∏k=1

(d+k (ζ ))σ3 − I

))12

, (77)

and ∫ x

+∞(|u(x′, t)|2 − 1) dx′

:= −i limζ→∞

ζ∈C\σ ′′OD

(ζ

(m8(ζ )(δ(ζ ))σ3

m−1∏k=1

(d+k (ζ ))σ3 − I

))11

. (78)


The analogue of Proposition 3.1 is

PROPOSITION A.1.1 ([38]). The solution of the RHP for m8(ζ ): C \ σ ′′OD →

M2(C) formulated in Lemma A.1.3 has the (integral equation) representation

m8(ζ ) = (I + ζ−1+8

0)P8(ζ )

(m

8

d(ζ )+∫σ ′′c

m8−(µ)(υ8(µ)− I)

(µ− ζ )

dµ

2π i

),

ζ ∈ C \ σ ′′OD ,

where

m8

d(ζ ) = I +N∑

n=m

(Res(m8(ζ ); ςn)

(ζ − ςn)+ σ1Res(m8(ζ ); ςn) σ1

(ζ − ςn)

),

v8(·) is a generic notation for the jump matrices of m8(ζ ) on σ ′′c (Lemma A.1.3(ii)),

and +8

0 and P 8(ζ ) are specified below. The solution of the above (integral) equa-tion can be written as the ordered factorisation

m8(ζ ) = (I + ζ−1+8

0)P8(ζ )m

8

d(ζ )mc(ζ ), ζ ∈ C \ σ ′′

OD ,

where m8

d(ζ ) = σ1m8

d(ζ ) σ1 (∈ SL(2,C)) has the representation given above,

P 8(ζ ) = σ1P 8(ζ ) σ1 is chosen so that +8

0 is idempotent, I + ζ−1+8

0 (∈ M2(C))

is holomorphic in a punctured neighbourhood of the origin, with +8

0 = σ1+8

0 σ1

(∈ GL(2,C)) such that det(I + ζ−1+8

0)|ζ=±1 = 0, and having the finite, order 2,matrix involutive structure

+8

0 =(

+8ei(k+1/2)π (1+ (+8)2)1/2e−iϑ8

(1 + (+8)2)1/2eiϑ8

+8e−i(k+1/2)π

), k ∈ Z,


where +8 and ϑ 8 are obtained from the relation

+8

0 = P 8(0)m8d(0)m

c(0)(δ(0))σ3

(m−1∏k=1

(d+k (0))σ3

)σ2,

and satisfying tr(+8

0) = 0, det(+8

0) = −1, and +8

0+8

0 = I, and mc(ζ ): C \σ ′′c → SL(2,C) solves the following RHP: (1) mc(ζ ) is piecewise (sectionally)

holomorphic ∀ζ ∈ C \ σ ′′c ;

(2) mc±(ζ ) := lim

ζ ′→ζ

ζ ′∈± side of σ ′′c

mc(ζ ′)

satisfy the jump condition mc+(ζ ) = mc−(ζ )υc(ζ ), ζ ∈ σ ′′c , where υc(ζ ) =

exp(−ik(ζ )(x + 2λ(ζ )t)ad(σ3))G8(ζ ), ζ ∈ R, with G8(ζ ) given in Lem-

ma A.1.3(ii), υc(ζ ) = I + CKn (ζ − ςn)

−1σ+, ζ ∈ Kn, and υc(ζ ) = I + CLn (ζ −

ςn)−1σ−, ζ ∈ Ln, n ∈ {1, 2, . . . , m− 1}, with CK

n and CLn given in Lemma A.1.3;

(3) mc(ζ ) = ζ→∞ζ∈C\σ ′′c

I +O(ζ−1); and (4) mc(ζ ) = σ1mc(ζ ) σ1.


LEMMA A.1.4. For m ∈ {1, 2, . . . , N}, set σd := ⋃Nn=m({ςn} ∪ {ςn}), and let

σc = {ζ ; Im(ζ ) = 0} with orientation from−∞ to+∞. Let X(ζ ): C\(σd∪σc)→M2(C) solve the following RHP:

(i) X(ζ ) is piecewise (sectionally) meromorphic ∀ζ ∈ C \ σc;(ii) X±(ζ ) := lim ζ ′→ζ

ζ ′∈± side of σc

X(ζ ′) satisfy the jump condition

X+(ζ ) = X−(ζ ) exp(−ik(ζ )(x + 2λ(ζ )t) ad(σ3))G8(ζ ), ζ ∈ R;

(iii) X(ζ ) has simple poles in σd with

Res(X(ζ ); ςn) = limζ→ςn

X(ζ )gn(δ(ςn))−2

(m−1∏k=1

(d+k (ςn))−2

)σ−,

n ∈ {m,m+ 1, . . . , N},Res(X(ζ ); ςn) = σ1Res(X(ζ ); ςn) σ1, n ∈ {m,m+ 1, . . . , N};

(iv) det(X(ζ ))|ζ=±1 = 0;(v) X(ζ ) =ζ→0 ζ

−1(δ(0))σ3(∏m−1

k=1 (d+k (0))

σ3)σ2 +O(1);(vi) X(ζ ) = ζ→∞


(vii) X(ζ ) = σ1X(ζ ) σ1 and X(ζ−1) = ζX(ζ )(δ(0))σ3(∏m−1

k=1 (d+k (0))

σ3)σ2.

Then, as t → −∞ and x → +∞ such that z0 := x/t < −2 and (x, t) ∈ �m,m8(ζ ): C \ σ ′′

OD → M2(C) has the following asymptotics:

m8(ζ ) = (I +O(F (ζ ) exp(−�|t|)))X(ζ ),

390 A. H. VARTANIAN

where � := 4 min m∈{1,2,...,N}n∈{1,2,...,m−1}

{sin(φn)|cos(φn) − cos(φm)|} (> 0), and, for i, j ∈{1, 2}, (F (ζ ))ij =ζ→∞ O(|ζ |−1) and (F (ζ ))ij =ζ→0 O(1). Furthermore, let


ζ∈C\(σd∪σc)

(ζ

(X(ζ )(δ(ζ ))σ3

m−1∏k=1

(d+k (ζ ))σ3 − I

))12

+

+O(exp(−�|t|)), (79)

and ∫ x

+∞(|u(x′, t)|2 − 1) dx′

:= −i limζ→∞

ζ∈C\(σd∪σc)

(ζ

(X(ζ )(δ(ζ ))σ3

m−1∏k=1

(d+k (ζ ))σ3 − I

))11

+

+O(exp(−�|t|)). (80)



LEMMA A.1.5. The solution of the RHP for X(ζ ): C \ (σd ∪ σc) → M2(C)

formulated in Lemma A.1.4 is given by the following ordered factorisation,

X(ζ ) = (I + ζ−1+0)P (ζ )md(ζ )Mc(ζ ), ζ ∈ C \ (σd ∪ σc),

where md(ζ ) = σ1md(ζ ) σ1 (∈ SL(2,C)) has the (series) representation

md(ζ ) = I +N∑

n=m

(Res(X(ζ ); ςn)

ζ − ςn+ σ1Res(X(ζ ); ςn)σ1

ζ − ςn

),

P (ζ ) = σ1P (ζ ) σ1 is chosen (see Lemma A.1.7 below) so that +0 is idempo-tent, I + ζ−1+0 is holomorphic in a punctured neighbourhood of the origin, with+0 = σ1+0 σ1 (∈ GL(2,C)) and det(I + ζ−1+0)|ζ=±1 = 0, and determined by+0 = P (0)md(0)Mc(0)(δ(0))σ3(

∏m−1k=1 (d

+k (0))

σ3)σ2, and satisfying tr(+0) = 0,det(+0) = −1, and +0+0 = I, and Mc(ζ ): C \ σc → SL(2,C) solves thefollowing RHP: (1) Mc(ζ ) is piecewise (sectionally) holomorphic ∀ζ ∈ C \ σc;(2) Mc±(ζ ) := lim ζ ′→ζ

±Im(ζ ′)>0Mc(ζ ′) satisfy, for ζ ∈ R, the jump condition

Mc+(ζ ) = Mc

−(ζ )e−ik(ζ )(x+2λ(ζ )t) ad(σ3)×

×((1 − r(ζ )r(ζ ))δ−(ζ )/δ+(ζ ) − r(ζ )

(δ−(ζ )δ+(ζ ))−1

∏m−1k=1 (d

+k (ζ ))

2

r(ζ )

δ−(ζ )δ+(ζ )∏m−1

k=1 (d+k (ζ ))

−2 δ+(ζ )/δ−(ζ )

);

(3) Mc(ζ ) = ζ→∞ζ∈C\σc

I +O(ζ−1); and (4) Mc(ζ ) = σ1Mc(ζ ) σ1.



LEMMA A.1.6. Let ε be an arbitrarily fixed, sufficiently small positive real num-ber, and, for z ∈ {λ1, λ2}, with λ1 and λ2 given in Theorem 2.2.1, Equation (10),set U(z; ε) := {ζ ; |ζ − z| < ε}. Then, as t → −∞ and x → +∞ suchthat z0 := x/t < −2, for ζ ∈ C \ ⋃z∈{λ1,λ2} U(z; ε), Mc(ζ ) has the followingasymptotics:

Mc11(ζ )

= 1 +O

((cS(λ1)c(λ2, λ3, λ3)√λ2(z

20 + 32) (ζ − λ1)

+ cS(λ2)c(λ1, λ3, λ3)√λ1(z

20 + 32) (ζ − λ2)

)ln|t|

(λ1 − λ2)t

),

Mc12(ζ )

= ei<−(0)

2

( √ν(λ1) λ

−2iν(λ1)1√|t|(λ1 − λ2) (z

20 + 32)1/4

(λ1ei(:−(z0,t )− 3π

4 )

(ζ − λ1)+ λ2e−i(:−(z0,t )− 3π

4 )

(ζ − λ2)

)+

+O

((cS(λ1)c(λ2, λ3, λ3)√λ2(z

20 + 32) (ζ − λ1)

+ cS(λ2)c(λ1, λ3, λ3)√λ1(z

20 + 32) (ζ − λ2)

)ln|t|

(λ1 − λ2)t

)),

Mc21(ζ )

= e−i<−(0)

2

( √ν(λ1)λ

2iν(λ1)

1√|t|(λ1 − λ2) (z20 + 32)1/4

(λ1e−i(:−(z0,t )− 3π

4 )

(ζ − λ1)+ λ2ei(:−(z0,t )− 3π

4 )

(ζ − λ2)

)+

+O

((cS(λ1)c(λ2, λ3, λ3)√λ2(z

20 + 32) (ζ − λ1)

+ cS(λ2)c(λ1, λ3, λ3)√λ1(z

20 + 32) (ζ − λ2)

)ln|t|

(λ1 − λ2)t

)),

Mc22(ζ )

= 1 +O

((cS(λ1)c(λ2, λ3, λ3)√λ2(z

20 + 32) (ζ − λ1)

+ cS(λ2)c(λ1, λ3, λ3)√λ1(z

20 + 32) (ζ − λ2)

)ln|t|

(λ1 − λ2)t

),

where λ3, ν(·), :−(z0, t), and <−(·), respectively, are given in Theorem 2.2.1,Equations (10), (11), (17), and (19), ‖(· − λk)

−1‖L∞(C \⋃z∈{λ1 ,λ2} U(z;ε)) < ∞, k ∈{1, 2}, Mc(ζ ) = σ1Mc(ζ ) σ1, and (Mc(0)σ2)

2 = I (+O(t−1 ln|t|)).Sketch of proof. Proceeding as in the proof of Lemma 6.1 in [38] and particu-

larising it to the case of the RHP for Mc(ζ ) stated in Lemma A.1.5, one arrivesat

Mc11(ζ ) = 1+ r(λ1)(δ

0B)

−2e− 3πν2 e

3π i4


21 XB

√|t|×

×∫ +∞

0

(e−


2e

3π i4 zDiν(z)

)ziνe−

z24 dz−

− r(λ1)(1 − |r(λ1)|2)−1(δ0B)

−2e−iπ4


21 e− πν2 XB

√|t|×

392 A. H. VARTANIAN

×∫ +∞

0

(e

iπ4 ∂zDiν(z)+ i

2e−

iπ4 zDiν(z)

)ziνe−

z24 dz+

+ r(λ1)(δ0A)

−2e− πν2 (−1)−iνe

iπ4


21 XA

√|t|×

×∫ +∞

0

(e−


2e

iπ4 zD−iν(z)

)z−iνe−

z24 dz−

− r(λ1)(1− |r(λ1)|2)−1(δ0A)

−2e−3π i

4


21 eπν2 (−1)iνXA

√|t|×

×∫ +∞

0

(e


2e−

3π i4 zD−iν(z)

)z−iνe−

z24 dz+

+O

((cS(λ1)c(λ2, λ3, λ3)(δ

0B)

−2

(ζ − λ1)|λ1 − λ3|√(λ1 − λ2) XB

+

+ cS(λ2)c(λ1, λ3, λ3)(δ0A)

−2

(ζ − λ2)|λ2 − λ3|√(λ1 − λ2) XA

)ln|t|t

),

Mc12(ζ ) =

(r(λ1)(1− |r(λ1)|2)−1(δ0

B)2e

iπ4

2π i(ζ − λ1)e−πν2 XB

√|t| − r(λ1)(δ0B)

2e−3πν

2 e−3π i

4

2π i(ζ − λ1)XB

√|t|)×

×∫ +∞

0D−iν(z)z

−iνe−z24 dz+

+(r(λ1)(1 − |r(λ1)|2)−1(δ0

A)2e

3π i4

2π i(ζ − λ2)eπν2 (−1)−iνXA

√|t|−

− r(λ1)(δ0A)

2e− πν2 e− iπ

4

2π i(ζ − λ2)(−1)−iνXA

√|t|)∫ +∞

0Diν(z)z

iνe−z24 dz+

+O

((cS(λ1)c(λ2, λ3, λ3)(δ

0B)

2

(ζ − λ1)|λ1 − λ3|√(λ1 − λ2) XB

+

+ cS(λ2)c(λ1, λ3, λ3)(δ0A)

2

(ζ − λ2)|λ2 − λ3|√(λ1 − λ2) XA

)ln |t|t

),

Mc21(ζ ) = −

(r(λ1)(1− |r(λ1)|2)−1(δ0

B)−2e− iπ

4

2π i(ζ − λ1)e−πν2 XB

√|t| − r(λ1)(δ0B)

−2e− 3πν2 e

3π i4

2π i(ζ − λ1)XB

√|t|)×

×∫ +∞

0Diν(z)z

iνe−z24 dz−

−(r(λ1)(1− |r(λ1)|2)−1(δ0

A)−2e− 3π i

4

2π i(ζ − λ2)eπν2 (−1)iνXA

√|t| −

− r(λ1)(δ0A)

−2e− πν2 e

iπ4

2π i(ζ − λ2)(−1)iνXA

√|t|)∫ +∞

0D−iν(z)z

−iνe−z24 dz+


+O

((cS(λ1)c(λ2, λ3, λ3)(δ

0B)

−2

(ζ − λ1)|λ1 − λ3|√(λ1 − λ2) XB

+

+ cS(λ2)c(λ1, λ3, λ3)(δ0A)

−2

(ζ − λ2)|λ2 − λ3|√(λ1 − λ2) XA

)ln |t|t

),

Mc22(ζ ) = 1− r(λ1)(δ

0B)

2e− 3πν2 e− 3π i

4


12 XB

√|t|×

×∫ +∞

0

(e


2e−

3π i4 zD−iν(z)

)z−iνe−

z24 dz+

+ r(λ1)(1− |r(λ1)|2)−1(δ0B)

2eiπ4


12 e−πν2 XB

√|t|×

×∫ +∞

0

(e−


2e

iπ4 zD−iν(z)

)z−iνe−

z24 dz−

− r(λ1)(δ0A)

2e−πν2 (−1)iνe−

iπ4


12 XA

√|t|×

×∫ +∞

0

(e

iπ4 ∂zDiν(z)+ i

2e−

iπ4 zDiν(z)

)ziνe−

z24 dz+

+ r(λ1)(1 − |r(λ1)|2)−1(δ0A)

2(−1)iνe3π i4


12 eπν2 XA

√|t|×

×∫ +∞

0

(e−


2e

3π i4 zDiν(z)

)ziνe−

z24 dz+

+O

((cS(λ1)c(λ2, λ3, λ3)(δ

0B)

2

(ζ − λ1)|λ1 − λ3|√(λ1 − λ2) XB

+

+ cS(λ2)c(λ1, λ3, λ3)(δ0A)

2

(ζ − λ2)|λ2 − λ3|√(λ1 − λ2) XA

)ln |t|t

),

where r(ζ ) = r(ζ )∏m−1

k=1 (d+k (ζ ))

−2 (|r(λ1)| = |r(λ1)|), ν = ν(λ1),

δ0B = |λ1 − λ3|iν(2|t|(λ1 − λ2)

3λ−31 )

iν2 eY(λ1)×

× exp

(− it

2(λ1 − λ2)(z0 + λ1 + λ2)

),

δ0A = |λ2 − λ3|−iν(2|t|(λ1 − λ2)

3λ−32 )−

iν2 eY(λ2)×

× exp

(it

2(λ1 − λ2)(z0 + λ1 + λ2)

),

Y(λ1) = i

2π

∫ λ2

0ln|µ− λ1| d ln(1 − |r(µ)|2)+

394 A. H. VARTANIAN

+ i

2π

∫ +∞

λ1

ln|µ− λ1| d ln(1− |r(µ)|2),

Y(λ2) = −Y(λ1)+ i

2π

∫ λ2

0ln|µ| d ln(1− |r(µ)|2)+

+ i

2π

∫ +∞

λ1

ln|µ| d ln(1− |r(µ)|2),

XB = XB, XA = XA, βIB0

12 = βIB0

21 =√

2π e−πν2 e

3π i4

r(λ1);(iν),

βIA0

12 = βIA0

21 =√

2π e− πν2 e− 3π i

4

r(λ1) ;(iν),

;(·) is the gamma function [51], and D∗(·) is the parabolic cylinder function [51].Proceeding, now, as at the end of the sketch of the proof of Lemma 4.2, oneobtains the result stated in the lemma. Furthermore, one shows that the symme-

try reduction Mc(ζ ) = σ1Mc(ζ ) σ1 is satisfied, and verifies that (Mc(0)σ2)2 =

I + O(t−1 ln|t|). ✷The analogue of Proposition 4.1 is

PROPOSITION A.1.2. For m ∈ {1, 2, . . . , N}, set Res(X(ζ ); ςn) :=(

an bn

cn dn

),

n ∈ {m,m + 1, . . . , N}. Then bn = −anMc12(ςn)/M

c22(ςn), dn = −cnM

c12(ςn)/

Mc22(ςn), and {an, cn}Nn=m satisfy the following (nonsingular) system of 2(N−m+1)

linear inhomogeneous algebraic equations,

A B

B A

am

am+1...

aN

cm

cm+1...

cN

=

gDmMc12(ςm)

gDm+1Mc12(ςm+1)...

gDNMc12(ςN)

gDmMc22(ςm)

gDm+1Mc22(ςm+1)...

gDNMc22(ςN)

,

where

Aij :=

det(Mc(ςi))+gDi W(Mc

12(ςi ),Mc22(ςi))

Mc22(ςi)

, i = j ∈ {m,m+ 1, . . . , N},

− gDi (Mc12(ςi)M

c22(ςj )−Mc

22(ςi )Mc12(ςj ))

(ςi−ςj )Mc22(ςj )

, i �= j ∈ {m,m+ 1, . . . , N},

Bij := −gDi (Mc22(ςi)M

c22(ςj )−Mc

12(ςi)Mc12(ςj ))

(ςi − ςj )Mc22(ςj )

,

i, j ∈ {m,m+ 1, . . . , N},


gDj = |gj |eiθgj exp(2ik(ςj )(x + 2λ(ςj )t))(δ(ςj))−2

m−1∏k=1

(d+k (ςj ))−2,

j ∈ {m,m+ 1, . . . , N},with |gj | and θgj given in Lemma 3.1(iii), and

W(Mc12(z),M

c22(z)) =

∣∣∣∣ Mc12(z) Mc

22(z)

∂zMc12(z) ∂zM

c22(z)

∣∣∣∣ .The analogue of Proposition 4.2 is

PROPOSITION A.1.3. As t → −∞ and x → +∞ such that z0 := x/t < −2and (x, t) ∈ �m, m ∈ {1, 2, . . . , N}, for n ∈ {m+ 1,m+ 2, . . . , N},

an = O(e−ג−|t |), bn = O(t−1/2(z20 + 32)−1/4e−ג−|t |),

cn = O(e−ג−|t |), dn = O(t−1/2(z20 + 32)−1/4e−ג−|t |),

where −ג := 4 min m∈{1,2,...,N}n∈{m+1,m+2,...,N}

{sin(φn)|cos(φn)− cos(φm)|} (> 0), and

am = a0m +

1√|t|a1m +O

(cS(z0)

(z20 + 32)1/2

ln|t|t

)=: gDmg

Dm(ςm − ςm)

−1

(1 + gDmgDm(ςm − ςm)

−2)+

+ 1√|t|(gDmg

Dm(ςm − ςm)

−1(gDm∂ζMc12(ςm)+ gDm∂ζM

c12(ςm))

(1+ gDmgDm(ςm − ςm)−2)2

+

+ gDmMc12(ςm)

(1+ gDmgDm(ςm − ςm)−2)

)+O

(cS(z0)

(z20 + 32)1/2

ln|t|t

),

bm = 1√|t|b1m +O

(cS(z0)

(z20 + 32)1/2

ln|t|t

)=: − 1√|t|

gDmgDm (ςm − ςm)

−1Mc12(ςm)

(1 + gDmgDm(ςm − ςm)

−2)+O

(cS(z0)

(z20 + 32)1/2

ln|t|t

),

cm = c0m +

1√|t| c1m +O

(cS(z0)

(z20 + 32)1/2

ln|t|t

)=: gDm

(1 + gDmgDm(ςm − ςm)−2)

+

+ 1√|t|(gDmg

Dm(ςm − ςm)

−1Mc12(ςm)− gDmg

Dm∂ζM

c12(ςm)

(1 + gDmgDm(ςm − ςm)−2)

+

+ gDm(gDm∂ζM

c12(ςm)+ gDm∂ζM

c12(ςm))

(1+ gDmgDm(ςm − ςm)

−2)2

)+O

(cS(z0)

(z20 + 32)1/2

ln|t|t

),

396 A. H. VARTANIAN

dm = 1√|t|d1m +O

(cS(z0)

(z20 + 32)1/2

ln|t|t

)=: − 1√|t|

gDmMc12(ςm)

(1+ gDmgDm(ςm − ςm)

−2)+O

(cS(z0)

(z20 + 32)1/2

ln|t|t

),

where

Mc12(ζ ) =

√ν(λ1) e

i<−(0)2 λ

−2iν(λ1)

1√(λ1 − λ2) (z

20 + 32)1/4

(λ1ei(:−(z0,t )− 3π

4 )

(ζ − λ1)+ λ2e−i(:−(z0,t )− 3π

4 )

(ζ − λ2)

),

with ν(·), λ1, λ2, λ3, <−(·), and :−(z0, t) specified in Lemma A.1.6, and cS(z0)

given in Proposition 4.2. Furthermore, setting

Y :=(

A B

B A

),

0 < | det(Y)|2 �N∏

j=m

(1+ sin2(φm)|γm|2P−2(φm, φk)Q

−2(φm)

sin2( 12 (φm + φj ))

e2φ(x,t)

)2

+O

(cS(z0)

(z20 + 32)1/2

ln|t|t

), (81)

where φ(x, t), P(φm, φk), and Q(φm) are defined in Equations (67), (68), and (69),respectively.

The analogue of Lemma 4.3 is (see, also, Remark 4.1)

LEMMA A.1.7. As t → −∞ and x → +∞ such that z0 := x/t < −2 and(x, t) ∈ �m, m ∈ {1, 2, . . . , N},

P (ζ ) = ζ+a−1

ζ+a−2a−3

ζ+a−4a−3

ζ+a−4ζ+a−1ζ+a−2

,

where

a−1 = a−2 = 1+∞∑p=1

p−1∑q=0

a1pq(z0)(ln|t|)q

|t|p/2+

+O(e−4|t |min m∈{1,2,...,N}

n∈{m+1,m+2,...,N}{sin(φn)|cos(φn)−cos(φm)|})

,

a−3 =∞∑p=1

p−1∑q=0

a3pq(z0)(ln|t|)q

|t|p/2+


+O(e−4|t |min m∈{1,2,...,N}


,

a−4 = 1 +∞∑p=1

p−1∑q=0

a4pq(z0)(ln|t|)q

|t|p/2+

+O(e−4|t |min m∈{1,2,...,N}


,

akpq(z0) ∈ cS(z0), k ∈ {1, 3, 4}, and P (ζ ) = σ1P (ζ ) σ1.


PROPOSITION A.1.4. Set a110(z0) =: a1, a2

10(z0) =: a2, a310(z0) =: a3, and

a410(z0) =: a4. Then as t → −∞ and x → +∞ such that z0 := x/t < −2

and (x, t) ∈ �m, m ∈ {1, 2, . . . , N},(+0)11

= − c0m

ςmiδ−1(0)e2i

∑m−1k=1 φk + iδ−1(0)e2i

∑m−1k=1 φk

√|t|(−(a1 − a2)

c0m

ςm−(

b1m

ςm+ c1

m

ςm

)+

+ a3

(1− a0

m

ςm

)−(

1 − a0m

ςm

)2δ(0)

√ν(λ1) cos(:−(z0, t)− 3π

4 )√(λ1 − λ2) (z

20 + 32)1/4

)+

+O

(cS(z0)

(z20 + 32)1/2

ln|t|t

),

(+0)12

= −(

1 − a0m

ςm

)iδ(0)e−2i

∑m−1k=1 φk + iδ(0)e−2i

∑m−1k=1 φk

√|t|(−(a1 − a2)

(1− a0

m

ςm

)+

+(

a1m

ςm+ d1

m

ςm

)+ a3

c0m

ςm− c0

m

ςm

2δ−1(0)√ν(λ1) cos(:−(z0, t)− 3π

4 )√(λ1 − λ2) (z

20 + 32)1/4

)+

+O

(cS(z0)

(z20 + 32)1/2

ln|t|t

),

(+0)21

=(

1 − a0m

ςm

)iδ−1(0)e2i

∑m−1k=1 φk + iδ−1(0)e2i

∑m−1k=1 φk

√|t|((a1 − a2)

(1 − a0

m

ςm

)−

−(

a1m

ςm+ d1

m

ςm

)− a3

c0m

ςm+ c0

m

ςm

2δ(0)√ν(λ1) cos(:−(z0, t)− 3π

4 )√(λ1 − λ2) (z

20 + 32)1/4

)+

+O

(cS(z0)

(z20 + 32)1/2

ln|t|t

),

(+0)22

= c0m

ςmiδ(0)e−2i

∑m−1k=1 φk + iδ(0)e−2i

∑m−1k=1 φk

√|t|((a1 − a2)

c0m

ςm+(

b1m

ςm+ c1

m

ςm

)−

398 A. H. VARTANIAN

− a3

(1− a0

m

ςm

)+(

1 − a0m

ςm

)2δ−1(0)

√ν(λ1) cos(:−(z0, t)− 3π

4 )√(λ1 − λ2) (z

20 + 32)1/4

)+

+O

(cS(z0)

(z20 + 32)1/2

ln|t|t

).


PROPOSITION A.1.5. Let φ(x, t), P(φm, φk), and Q(φm) be defined by Equa-tions (67), (68), and (69), respectively. Then, for θγm = ±π/2, as t → −∞ andx →+∞ such that z0 := x/t < −2 and (x, t) ∈ �m, m ∈ {1, 2, . . . , N},a

0m

= −2i sin(φm)|γm|2P−2(φm, φk)Q−2(φm)e2φ(x,t)

(1 − |γm|2P−2(φm, φk)Q−2(φm)e2φ(x,t)),

c0m

= ∓2 sin(φm)|γm|δ−1(0)ei(φm+s−)+φ(x,t)P−1(φm, φk)Q−1(φm)

(1− |γm|2P−2(φm, φk)Q−2(φm)e2φ(x,t)),

a1m

= ∓ 16iλ21 sin2(φm)|γm|3√ν(λ1) P

−3(φm, φk)Q−3(φm) cos(s−)e3φ(x,t)

(1 − |γm|2P−2(φm, φk)Q−2(φm)e2φ(x,t))2(λ21 − 2λ1 cos(φm)+ 1)2

√(λ1 − λ2) (z

20 + 32)1/4

×

×(((λ1 + λ2) cos(φm)− 2) cos

(:−(z0, t)− 3π

4

)−

− (λ1 − λ2) sin(φm) sin

(:−(z0, t)− 3π

4

))∓

∓ 2λ1 sin(φm)|γm|√ν(λ1) P−1(φm, φk)Q

−1(φm)eφ(x,t)

(1 − |γm|2P−2(φm, φk)Q−2(φm)e2φ(x,t))(λ21 − 2λ1 cos(φm)+ 1)

√(λ1 − λ2) (z

20 + 32)1/4

×

×(

2 cos(s−) cos

(:−(z0, t)− 3π

4

)− (λ1 + λ2) cos(φm + s−)×

× cos

(:−(z0, t)− 3π

4

)+ (λ1 − λ2) sin(φm + s−) sin

(:−(z0, t)− 3π

4

)+

+ 2i sin(s−) cos

(:−(z0, t)− 3π

4

)− i(λ1 + λ2) sin(φm + s−)×

× cos

(:−(z0, t)− 3π

4

)− i(λ1 − λ2) cos(φm+ s−) sin

(:−(z0, t)− 3π

4

)),

b1m

= 2iλ1 sin(φm)|γm|2√ν(λ1) δ(0)e−i(φm+s−)+2φ(x,t)P−2(φm, φk)Q−2(φm)


√(λ1 − λ2) (z

20 + 32)1/4

×

×(

2 cos(s−) cos

(:−(z0, t)− 3π

4

)− (λ1 + λ2) cos(φm + s−)×


× cos

(:−(z0, t)− 3π

4

)+ (λ1 − λ2) sin(φm + s−) sin

(:−(z0, t)− 3π

4

)+

+ 2i sin(s−) cos

(:−(z0, t)− 3π

4

)− i(λ1 + λ2) sin(φm + s−)×

× cos

(:−(z0, t)− 3π

4


(:−(z0, t)− 3π

4

)),

c1m

= − 16λ21 sin2(φm)|γm|2√ν(λ1) δ

−1(0)ei(φm+s−)+2φ(x,t)P−2(φm, φk)Q−2(φm) cos(s−)


√(λ1 − λ2) (z

20 + 32)1/4

×

×(((λ1 + λ2) cos(φm)− 2) cos

(:−(z0, t)− 3π

4

)−


(:−(z0, t)− 3π

4

))−

− 2iλ1 sin(φm)|γm|2√ν(λ1) δ−1(0)ei(φm+s−)+2φ(x,t)P−2(φm, φk)Q

−2(φm)


√(λ1 − λ2) (z

20 + 32)1/4

×

×(

2 cos(s−) cos

(:−(z0, t)− 3π

4

)− (λ1 + λ2) cos(φm + s−)×

× cos

(:−(z0, t)− 3π

4

)+ (λ1 − λ2) sin(φm + s+) sin

(:−(z0, t)− 3π

4

)−

− 2i sin(s−) cos

(:−(z0, t)− 3π

4

)+ i(λ1 + λ2) sin(φm + s−)×

× cos

(:−(z0, t)− 3π

4

)+ i(λ1 − λ2)cos(φm+ s−)sin

(:−(z0, t)− 3π

4

))+

+ 8λ21 sin2(φm)|γm|2√ν(λ1) δ

−1(0)ei(φm+s−)+2φ(x,t)P−2(φm, φk)Q−2(φm)

(1 − |γm|2P−2(φm, φk)Q−2(φm)e2φ(x,t))(λ2

1 − 2λ1 cos(φm)+ 1)2√(λ1 − λ2) (z

20 + 32)1/4

×

×((

((λ1 + λ2) cos(φm)− 2) cos

(:−(z0, t)− 3π

4

)−


(:−(z0, t)− 3π

4

))cos(s−)−

− i

(((λ1 + λ2) cos(φm)− 2) cos

(:−(z0, t)− 3π

4

)−


(:−(z0, t)− 3π

4

))sin(s−)

),

d1m

= ± 2λ1 sin(φm)|γm|√ν(λ1) P

−1(φm, φk)Q−1(φm)eφ(x,t)


√(λ1 − λ2) (z

20 + 32)1/4

×

×(

2 cos(s−) cos

(:−(z0, t)− 3π

4

)− (λ1 + λ2) cos(φm + s−)×

400 A. H. VARTANIAN

× cos

(:−(z0, t)− 3π

4

)+ (λ1 − λ2) sin(φm + s−) sin

(:−(z0, t)− 3π

4

)+

+ 2i sin(s−) cos

(:−(z0, t)− 3π

4

)− i(λ1 + λ2) sin(φm + s−)×

× cos

(:−(z0, t)− 3π

4


(:−(z0, t)− 3π

4

)),

where s− is given in Theorem 2.2.1, Equation (11).


PROPOSITION A.1.6. As t → −∞ and x → +∞ such that z0 := x/t < −2and (x, t) ∈ �m, m ∈ {1, 2, . . . , N},

u(x, t) = i

((+0)12 + a−3 + bm + cm+

+√ν(λ1) e

i<−(0)2 λ

−2iν(λ1)

1√|t|(λ1 − λ2) (z20 + 32)1/4

(λ1ei(:−(z0,t )− 3π4 )+

+ λ2e−i(:−(z0,t )− 3π4 ))

)+ O

(cS(z0)

(z20 + 32)1/2

ln|t|t

), (82)∫ x

+∞(|u(x′, t)|2 − 1) dx′

= −i

((+0)11 + a−1 − a−2 + am + dm + 2i

m−1∑k=1

sin(φk)+

+ i

(∫ λ2

0+∫ +∞

λ1

)ln(1− |r(µ)|2) dµ

2π

)+

+O

(cS(z0)

(z20 + 32)1/2

ln|t|t

), (83)∫ x

−∞(|u(x′, t)|2 − 1) dx′ =

∫ x

+∞(|u(x′, t)|2 − 1) dx′ − 2

N∑n=1

sin(φn)−

−∫ +∞

−∞ln(1 − |r(µ)|2) dµ

2π. (84)


PROPOSITION A.1.7. As t → −∞ and x → +∞ such that z0 := x/t < −2and (x, t) ∈ �m, m ∈ {1, 2, . . . , N}, for θγm = ±π/2,

(+0)11

= i

(± 2 sin(φm)|γm|P−1(φm, φk)Q

−1(φm)eφ(x,t)

(1 − |γm|2P−2(φm, φk)Q−2(φm)e2φ(x,t))+


+√ν(λ1)√|t|(λ1 − λ2) (z

20 + 32)1/4

(− 2 cos

(:−(z0, t)− 3π

4

)cos(s−)+

+ 4 sin(φm)|γm|2P−2(φm, φk)Q−2(φm) sin(s− − φm) cos(:−(z0, t)− 3π

4 )e2φ(x,t)

(1 − |γm|2P−2(φm, φk)Q−2(φm)e2φ(x,t))+

+ 8λ21 sin2(φm)|γm|2P−2(φm, φk)Q

−2(φm)(1+ |γm|2P−2(φm, φk)Q−2(φm)e2φ(x,t)) cos(s−)e2φ(x,t)

(1 − |γm|2P−2(φm, φk)Q−2(φm)e2φ(x,t))2(λ21 − 2λ1cos(φm)+ 1)2

×

×(((λ1 + λ2) cos(φm)− 2) cos

(:−(z0, t)− 3π

4

)−


(:−(z0, t)− 3π

4

))+

+ 4λ1 sin(φm) cos(φm)|γm|2P−2(φm, φk)Q−2(φm)e2φ(x,t)


×

×(

2 cos

(:−(z0, t)− 3π

4

)sin(s− − φm)− (λ1 + λ2) sin(s−)×

× cos

(:−(z0, t)− 3π

4

)− (λ1 − λ2) cos(s−) sin

(:−(z0, t)− 3π

4

))))+

+O

(cS(z0)

(z20 + 32)1/2

ln|t|t

),

(+0)12

= −ie−i(θ−(1)+s−) + 2 sin(φm)|γm|2P−2(φm, φk)Q−2(φm)e−i(θ− (1)+φm+s−)+2φ(x,t)

(1 − |γm|2P−2(φm, φk)Q−2(φm)e2φ(x,t))+

+ 1√t

(2i Im(a1 − a2) sin(φm)|γm|2P−2(φm, φk)Q

−2(φm)e−i(θ−(1)+φm+s−)+2φ(x,t)

(1 − |γm|2P−2(φm, φk)Q−2(φm)e2φ(x,t))±

± 4i sin(φm)|γm|P−1(φm, φk)Q−1(φm)

√ν(λ1) e−i(θ− (1)+2s− )+φ(x,t) cos(:−(z0, t)− 3π

4 )

(1 − |γm|2P−2(φm, φk)Q−2(φm)e2φ(x,t))√(λ1 − λ2) (z

20 + 32)1/4

∓

∓ 16iλ21 sin2(φm)|γm|3P−3(φm, φk)Q

−3(φm)√ν(λ1) e−i(θ−(1)+s−)+3φ(x,t) cos(s−)


√(λ1 − λ2) (z

20 + 32)1/4

×

×(((λ1 + λ2) cos(φm)− 2) sin(φm) cos

(:−(z0, t)− 3π

4

)−

− (λ1 − λ2) sin2(φm) sin

(:−(z0, t)− 3π

4

)+

+ i

(((λ1 + λ2) cos(φm)− 2) cos(φm) cos

(:−(z0, t)− 3π

4

)−

− (λ1 − λ2) sin(φm) cos(φm) sin

(:−(z0, t)− 3π

4

))+

+ Im(a1 − a2)e−i(θ−(1)+s−)−

− 4λ1 sin(φm)|γm|P−1(φm, φk)Q−1(φm)

√ν(λ1) e−i(θ−(1)+s− )+φ(x,t)


√(λ1 − λ2) (z

20 + 32)1/4

×

402 A. H. VARTANIAN

×(∓ 2 sin(s− − φm) cos

(:−(z0, t)− 3π

4

)± (λ1 + λ2) sin(s−)×

× cos

(:−(z0, t)− 3π

4

)± (λ1 − λ2) cos(s−) sin

(:−(z0, t)− 3π

4

)))+

+O

(cS(z0)

(z20 + 32)1/2

ln|t|t

),

Im(a1 − a2)

= ±√ν(λ1) sin(s−) cos(:−(z0, t)− 3π

4 )(1 − |γm|2P−2(φm, φk)Q−2(φm)e2φ(x,t))√

(λ1 − λ2) (z20 + 32)1/4 sin(φm)|γm|P−1(φm, φk)Q−1(φm)eφ(x,t)

±

± 4λ21

√ν(λ1) sin(φm)|γm|P−1(φm, φk)Q

−1(φm) sin(s−)eφ(x,t)

(λ21 − 2λ1 cos(φm)+ 1)2

√(λ1 − λ2) (z

20 + 32)1/4

×

×(((λ1 + λ2) cos(φm)− 2) cos

(:−(z0, t)− 3π

4

)−


(:−(z0, t)− 3π

4

))±

± 2λ1√ν(λ1) cos(φm)|γm|P−1(φm, φk)Q

−1(φm)eφ(x,t)

(λ21 − 2λ1 cos(φm)+ 1)

√(λ1 − λ2) (z

20 + 32)1/4

×

×(

2 cos(s− − φm) cos

(:−(z0, t)− 3π

4

)−

− (λ1 + λ2) cos(s−) cos

(:−(z0, t)− 3π

4

)+

+ (λ1 − λ2) sin(s−) sin

(:−(z0, t)− 3π

4

))±

± 2√ν(λ1)|γm|P−1(φm, φk)Q

−1(φm) cos(s− − φm) cos(:−(z0, t)− 3π4 )eφ(x,t)√

(λ1 − λ2) (z20 + 32)1/4

+

+O

(cS(z0)

(z20 + 32)1/2

ln|t|t

),

Re(a1 − a2) = Re(a3) = Im(a3) = 0,

where θ−(·) is given in Theorem 2.2.1, Equation (9).


LEMMA A.1.8. As t → −∞ and x → +∞ such that z0 := x/t < −2 and(x, t) ∈ �m, m ∈ {1, 2, . . . , N}, u(x, t), the solution of the Cauchy problemfor the Df NLSE, and

∫ x

±∞(|u(x′, t)|2 − 1) dx′ have the leading-order asymptoticexpansions (for the lower sign) stated in Theorem 2.2.1, Equations (7)–(20).


Appendix B

In order to obtain the results of Theorems 2.2.3 and 2.2.4, the following Lemma,which is the analogue of Lemmae 4.2 and A.1.6, is requisite.

LEMMA B.1.1. Let ε be an arbitrarily fixed, sufficiently small positive real num-ber, and, for λ ∈ J := {(s1)

±1, (s2)±1}, where

s1 = −1

2(a1 − i(4 − a2

1)1/2) = eiϕ1,

ϕ1 := arctan

((4− a2

1)1/2

|a1|)∈(

0,π

2

), a1 < 0, |a1| < 2,

s2 = −1

2(a2 − i(4 − a2

2)1/2) = eiϕ2,

ϕ2 := − arctan

((4− a2

2)1/2

|a2|)∈(π

2, π

), a2 > 0, |a2| < 2,

with a1 and a2 given in Theorem 2.2.1, Equation (10), set U(λ; ε) := {z; |z− λ| <ε}. Then, for r(s1) = exp(−iε1π/2)|r(s1)|, ε1 ∈ {±1}, r(s2) = exp(iε2π/2)|r(s2)|,ε2 ∈ {±1}, 0 < r(s2)r(s2) < 1, and ζ ∈ C \ ⋃λ∈J

U(λ; ε), as t → +∞ andx →−∞ such that z0 := x/t ∈ (−2, 0), mc(ζ ) has the following asymptotics,

mc11(ζ ) = 1 +O

((c(z0)

(ζ − s1)+ c(z0)

(ζ − s2)

)e−4αt

βt

),

mc12(ζ ) =

ε1e−(

2a0t+sin(ϕ1)∫ 0−∞

ln(1−|r(µ)|2)(µ−cos ϕ1)

2+sin2 ϕ1

dµπ

)e−i(ϕ1+

∫ 0−∞

(µ−cos ϕ1) ln(1−|r(µ)|2)(µ−cos ϕ1)

2+sin2 ϕ1

dµπ

)2(|r(s1)|)−1(b0t)1/2(ζ − s1)

+

+ ε2e−(

2a0t−sin(ϕ3)∫ 0−∞

ln(1−|r(µ)|2)(µ−cos ϕ3)

2+sin2 ϕ3

dµπ

)e

i(ϕ3−

∫ 0−∞


2+sin2 ϕ3

dµπ

)2(|r(s2)|)−1(1 − r(s2)r(s2))(b0t)1/2(ζ − s2)

+

+O

((c(z0)

(ζ − s1)+ c(z0)

(ζ − s2)

)e−4αt

βt

),

mc21(ζ ) =

ε1e−(

2a0t+sin(ϕ1)∫ 0−∞

ln(1−|r(µ)|2)(µ−cos ϕ1)

2+sin2 ϕ1

dµπ

)e

i(ϕ1+

∫ 0−∞


2+sin2 ϕ1

dµπ

)2(|r(s1)|)−1(b0t)1/2(ζ − s1)

+

+ ε2e−(

2a0t−sin(ϕ3)∫ 0−∞

ln(1−|r(µ)|2)(µ−cos ϕ3)

2+sin2 ϕ3

dµπ

)e−i(ϕ3−

∫ 0−∞


2+sin2 ϕ3

dµπ

)2(|r(s2)|)−1(1− r(s2)r(s2))(b0t)1/2(ζ − s2)

+

+O

((c(z0)

(ζ − s1)+ c(z0)

(ζ − s2)

)e−4αt

βt

),

mc22(ζ ) = 1 +O

((c(z0)

(ζ − s1)+ c(z0)

(ζ − s2)

)e−4αt

βt

),

404 A. H. VARTANIAN

where

a0 = 1

2(z0 − a1)(4− a2

1)1/2 (> 0),

a0 = −1

2(z0 − a2)(4− a2

2)1/2 (> 0),

b0 = 1

2(z2

0 + 32)1/2(4 − a21)

1/2 (> 0),

b0 = 1

2(z2

0 + 32)1/2(4 − a22)

1/2 (> 0),

α := min{a0, a0}, β := min{b0, b0},and, for r(s1) = exp(iε1π/2)|r(s1)|, ε1 ∈ {±1}, r(s2) = exp(−iε2π/2)|r(s2)|,ε2 ∈ {±1}, 0 < r(s1)r(s1) < 1, and ζ ∈ C \ ⋃λ∈J

U(λ; ε), as t → −∞ andx →+∞ such that z0 ∈ (−2, 0),

mc11(ζ ) = 1+O

((c(z0)

(ζ − s1)+ c(z0)

(ζ − s2)

)e−4α|t |

βt

),

mc12(ζ ) = − ε1e

−(

2a0 |t |−sin(ϕ1)∫ +∞

0ln(1−|r(µ)|2)

(µ−cos ϕ1)2+sin2 ϕ1

dµπ

)e

i

(ϕ1−

∫+∞0

(µ−cos ϕ1) ln(1−|r(µ)|2 )(µ−cos ϕ1 )

2+sin2 ϕ1

dµπ

)2(|r(s1)|)−1(1 − r(s1)r(s1))(b0|t |)1/2(ζ − s1)

−

− ε2e−(

2a0 |t |+sin(ϕ3)∫ +∞

0ln(1−|r(µ)|2)


dµπ

)e−i

(ϕ3+

∫+∞0

(µ−cos ϕ3) ln(1−|r(µ)|2 )(µ−cos ϕ3 )

2+sin2 ϕ3

dµπ

)2(|r(s2)|)−1(b0|t |)1/2(ζ − s2)

+

+O

((c(z0)

(ζ − s1)+ c(z0)

(ζ − s2)

)e−4α|t |

βt

),

mc21(ζ ) = − ε1e

−(

2a0 |t |−sin(ϕ1)∫ +∞

0ln(1−|r(µ)|2)


dµπ

)e−i

(ϕ1−

∫ +∞0

(µ−cos ϕ1) ln(1−|r(µ)|2 )(µ−cos ϕ1)

2+sin2 ϕ1

dµπ

)2(|r(s1)|)−1(1 − r(s1)r(s1))(b0|t |)1/2(ζ − s1)

−

− ε2e−(

2a0 |t |+sin(ϕ3)∫ +∞

0ln(1−|r(µ)|2)


dµπ

)e

i

(ϕ3+

∫+∞0

(µ−cos ϕ3 ) ln(1−|r(µ)|2)(µ−cos ϕ3)

2+sin2 ϕ3

dµπ

)2(|r(s2)|)−1(b0|t |)1/2(ζ − s2)

+

+O

((c(z0)

(ζ − s1)+ c(z0)

(ζ − s2)

)e−4α|t |

βt

),

mc22(ζ ) = 1+O

((c(z0)

(ζ − s1)+ c(z0)

(ζ − s2)

)e−4α|t |

βt

),

where supζ∈C \⋃λ∈JU(λ;ε) |(ζ − (sn)

±1)−1| <∞, and mc(ζ ) = σ1mc(ζ) σ1.

Appendix C. Matrix Riemann–Hilbert Theory in the L2 Sobolev Space

In this Appendix, the theoretical foundation for this paper is presented. Beginningfrom the Lax-pair isospectral deformation formulation for a completely integrable


NLEE, in the sense of the ISM, a succinct review of several basic and key facts fromthe 2 × 2 matrix RH factorisation theory on unbounded self-intersecting contoursis presented: for complete details and proofs, see [40, 44–47, 50]. For simplicity,one begins with the solitonless sector, σd ≡ ∅, leading to the so-called ‘regular’RHP: inclusion of the (nonempty and finitely denumerable) discrete spectrum, σd ,is known as the ‘singular’ RHP, and is discussed below Theorem C.1.4.

For a completely integrable system of NLEEs, in the sense of the ISM, writethe spatial part of the associated Lax pair (see, for example, Proposition 2.1.1) as∂x$(x, t;λ) = (J (λ)+ R(x, t;λ))$(x, t;λ), where (x, t) ∈ R×[−T , T ], λ ∈ C,J (λ) := diag(z1(λ), z2(λ)) is rational with distinct entries, and R(x, t;λ) is off-diagonal. The orders of the poles of J (λ) and R(x, t;λ) must satisfy the followingrequirements (denote by PJ the set of poles of J (λ), and let k(λ′) denote the orderof the pole of λ′ ∈ PJ ): (1) every pole of R(x, t;λ) is a pole of J (λ); (2) if ∞ isa pole of J (λ) of order k(∞), then it is a pole of R(x, t;λ) of order not greaterthan k(∞) − 1; and (3) if λ′ is a finite pole of J (λ) of order k(λ′), then it is apole of R(x, t;λ) of order not greater than k(λ′). Hence, one has the followingrepresentations for J (λ) and R(x, t;λ): (1)

J (λ) =∑

λ′∈ PJ \{∞}

k(λ′)∑j=1

Jλ′,j (λ− λ′)−j +k(∞)∑l=0

J∞,lλl,

where Jλ′,j and J∞,l are M2(C)-valued, diagonal matrices with distinct elements;and (2)

R(x, t;λ) =∑

λ′∈ PJ \{∞}

k(λ′)∑j=1

rλ′,j (x, t)(λ − λ′)−j +k(∞)−1∑l=0

r∞,l(x, t)λl .

Remark C.1.1. Hereafter, for economy of notation, all explicit x, t dependen-cies are suppressed.

Denote by > the closure of {λ ∈ C; Re(z1(λ) − z2(λ)) = 0}. Decompose> into a finite union of piecewise smooth, simple, closed curves, > := ⋃

l∈L >l

(card(L) <∞). Denote by V the set of all self-intersections of >, V := {λ; >l ∩>m �= ∅, l �= m ∈ {1, 2, . . . , card(L)}} (it is assumed throughout that card(V) <

∞). Divide the complement of > into two disjoint open subsets of C, �+ and �−,each of which have finitely many components, �± := ⋃

l±∈L± �±l± (card(L±) <

∞), such that > admits an orientation so that it can be viewed either as a positively(counter-clockwise) oriented boundary, >+, for �+, or as a negatively (clockwise)oriented boundary, >−, for �−; moreover, for each component �±

l± , ∂�±l± has no

self-intersections.

DEFINITION C.1.1. For an M2(C)-valued function, f (λ), say, denote by f±(λ),respectively, the nontangential limits, if they exist, of f (λ) taken from �±. For

406 A. H. VARTANIAN

f (λ): >→ M2(C), define f (0)(λ) := f (λ), and, for k ∈ Z�1, f (j)(λ) := ∂j

λf (λ),j ∈ {1, 2, . . . , k}. For the piecewise smooth simple closed curve > = ⋃

l∈L >l ,and k ∈ Z�1, define the L2

M2(C)(>) Sobolev space Hk(>,M2(C)) as the setof all M2(C)-valued functions on > satisfying: (1) for l ∈ {1, 2, . . . , card(L)},f (j)�>l

, j ∈ {0, 1, . . . , k − 1}, exist pointwise and ∈ L2M2(C)(>l); and (2) for

l ∈ {1, 2, . . . , card(L)}, f (k−1)�>lis locally absolutely continuous and f (k)�>l

∈L2

M2(C)(>l). For k = 0, denote H 0(>,M2(C)) by L2M2(C)(>). Define

-Hk(>±,M2(C)) := {f : > → M2(C); f �∂�±

l±∈ Hk(∂�±

l±,M2(C)),

l± ∈ {1, 2, . . . , card(L±)}, k ∈ Z�1}: the norm on Hk(>±,M2(C)), k ∈ Z�1, isdefined as ‖f (·)‖Hk(>,M2(C)) := ‖f (·)‖2,k := (

∑l∈L∑k

j=0 ‖f (j)(·)‖2L2

M2(C)(>l))1/2.

With this norm, Hk(>±,M2(C)) is a Hilbert space: for k = 0, ‖f (·)‖2,0 =(∑

l∈L ‖f (·)‖2L2

M2(C)(>l)

)1/2.

The Cauchy integral operators on L2M2(C)(>) are defined as

(C±f )(λ) := limλ′→λλ′∈�±

∫>

f (z)

(z− λ′)dz

2π i:

note that C+−C− = id, where id is the identity operator on L2M2(C)(>). Since > =⋃

l∈L >l , where >l , l ∈ {1, 2, . . . , card(L)}, are piecewise smooth and simple, theCauchy integral operators are bounded from L2

M2(C)(>) into L2M2(C)(>); moreover,

the aforementioned orientation for >, that is, >±, provides the Cauchy integraloperators on L2

M2(C)(>) with the crucial property that ±C± are complementaryprojections, that is, C2+ = C+, C2− = −C−, C+C− = C−C+ = 0, where 0 is thenull operator on L2

M2(C)(>). Even though C± are not bounded in operator normon Hk(>,M2(C)), C± are bounded on

⊕α∈{±}H

k(>α,M2(C)); moreover, injec-tively, C±: Hk(>±,M2(C)) → Hk(>±,M2(C)), and C±: Hk(>∓,M2(C)) →H k(>,M2(C)) := ⋂

α∈{±}Hk(>α,M2(C)). Since, in the ISM, > is (usually) un-

bounded, the function f �>±= I �∈ Hk(>±,M2(C)), k ∈ Z�0; hence, for D ∈{>, >±}, embed Hk(D,M2(C)), k ∈ Z�0, into a larger Hilbert spaceHk

I (D,M2(C)) consisting of M2(C)-valued functions f (λ) on > ∪ (⋃

α∈{±} �α)

with the limit f (∞) at ∞ such that f (λ)−f (∞) ∈ Hk(D,M2(C)), with the normdefined by ‖f (·)||Hk

I (D,M2(C)) := ‖f (·)‖I,2,k := (|f (∞)|2 +‖f (·)− f (∞)‖22,k)

1/2.

HkI (D,M2(C)), k ∈ Z�0, is isomorphic to the Hilbert space direct sum of M2(C)

and Hk(D,M2(C)) (H kI (D,M2(C)) ≈ M2(C)⊕Hk(D,M2(C))).

Define: (1) GHkI (>

±,M2(C)) := {f (λ) ∈ HkI (>

±,M2(C)); det(f (λ)) �≡ 0};and (2) SHk

I (>±,M2(C)) := {f (λ) ∈ Hk

I (>±,M2(C)); det(f (λ)) = 1}. If

χc±(λ)− χc±(∞) ∈ ranC± (⊂ Hk(>±,M2(C))), where χc±(∞) := lim λ→∞λ∈�±

χc(λ),

denote by χc(λ) the sectionally holomorphic function on⋃

α∈{±} �α with bound-

ary values χc±(λ). Define: (1) H k(C \ >,M2(C)) := {χc(λ); χc±(λ) − χc±(∞) ∈


ranC±}; (2) GH k(C \ >,M2(C)) := {χc(λ) ∈ H k(C \ >,M2(C)); det(χc(λ)) �≡0}; and (3) SH k(C\>,M2(C)) := {χc(λ) ∈ H k(C\>,M2(C)); det(χc(λ)) = 1}.THEOREM C.1.1. Every v(λ) ∈ GHk

I (>−,M2(C)) ∗ GHk

I (>+,M2(C)) (A ∗

B := {xy; x ∈ A, y ∈ B}), λ ∈ >, admits an RH factorisation, v(λ) =(χc−(λ))−1�(λ)χc+(λ), where

�(λ) := diag

((λ− λ+λ− λ−

)k1

,

(λ− λ+λ− λ−

)k2), λ± ∈ �±,

and χc(λ) ∈ GH k(C \ >,M2(C)) (ki , i ∈ {1, 2}, are called the partial in-dices (uniquely determined by v(·) up to a permutation) of v(λ)); moreover, ifdet(v(λ)) = 1, χc(λ) can be chosen to be in SH k(C \ >,M2(C)), and

∑2j=1 kj =

0. The matrix χc(λ) ∈ HjI (>,M2(C)), for some j ∈ {0, 1, . . . , k}, k ∈ Z�1, is

said to be a solution of the RH factorisation problem of v(λ) if χc±(λ)− χc±(∞) ∈ranC± ⊂ Hk(>±,M2(C)). When v(∞) = I and �(λ) = I, χc±(λ) can be uniquelydetermined by letting χc±(∞) = I (canonical normalisation), in which case, χc±(λ),or χc(λ) (χc(∞) = I), is called the fundamental solution of the RHP of v(λ). Forthe ISM, v(∞) = I. Conversely, if v(λ) admits a factorisation v(λ) =(χc−(λ))−1�(λ)χc+(λ), then v(λ) ∈ GHk

I (>−,M2(C)) ∗GHk

I (>+,M2(C)).

PROPOSITION C.1.1. tr(R(λ)) = 0 ⇒ det(χc(λ)) = const.

DEFINITION C.1.2. A linear operator L on HkI (D,M2(C)) is Fredholm if: (1)

the complement of ran L is open in HkI (D;M2(C)); and (2) dim ker(L) and

dim coker(L) are finite. For L linear and Fredholm, i(L) := dim ker(L) −dim coker(L) is called the (Fredholm) index of L.

THEOREM C.1.2. Let k ∈ Z�1. If v(λ) in Theorem C.1.1 can be represented asthe following (algebraic) block triangular factorisation, v(λ) := (I−w−(λ))−1(I+w+(λ)), λ ∈ >, where w±(λ) ∈ Hk(>±,M2(C)), I±w±(λ) ∈ GHk

I (>±,M2(C)),

and w±(λ) are nilpotent, with degree of nilpotency 2, and if, as a linear operator onH k

I (>,M2(C)) :=⋂α∈{±}HkI (>

α,M2(C)),Cw: H kI (>,M2(C))→ H k

I (>,M2(C))

is defined as (f ∈ H kI (>,M2(C))) f �→ C+(fw−) + C−(f w+), then id − Cw,

where id is the identity operator on H kI (>,M2(C)), is Fredholm, that is, i(id −

Cw) = dim ker(id−Cw)−dim coker(id−Cw) = 0, dim ker(id−Cw) = 2∑

kj>0 kj ,and dim coker(id−Cw) = −2

∑kj<0 kj , where ki , i ∈ {1, 2}, are the partial indices

of v(λ); moreover, i(id − Cw) = 2 ind det(v(λ)) = 1π

∫>

d(arg det(v(·))) = 0,

where ind det(v(λ)), the index of det(v(λ)), equals∑2

j=1 kj . Define χc0 (λ) :=

((id − Cw)−1I)(λ): then the boundary values χc±(λ) := χc

0 (λ)(I ± w±(λ)) ∈(I + ranC±) ∩ GHk

I (>±,M2(C)) ⊂ (I + Hk(>±,M2(C))) ∩ GHk

I (>±,M2(C))

give the fundamental solution of the RH factorisation problem for v(λ).

408 A. H. VARTANIAN

THEOREM C.1.3. If all the partial indices of v(λ) are zero (ki = 0, i ∈ {1, 2}),then the Fredholm operator id − Cw is invertible on H k

I (>,M2(C)), namely,ker(id − Cw) = ∅ (dim ker(id − Cw) = 0).

LEMMA C.1.1. The RHP of v(λ) := (I − w−(λ))−1(I + w+(λ)) =(χc−(λ))−1χc+(λ), λ ∈ >, where w±(λ) ∈ Hk(>±,M2(C)), has a fundamentalsolution (χc(∞) = I, χc(λ) �≡ 0) only if 1

2π

∫>

d(arg det(v(·))) = 0. Conversely,if χc(λ) ∈ H k

I (>,M2(C)), k ∈ Z�1, χc(∞) = I is a solution of the RHP ofv(λ) on >, and 1

2π

∫>

d(arg det(v(·))) = 0, then χc(λ) is a fundamental solution;furthermore, det(v(λ)) = 1 ⇒ det(χc(λ)) = 1.

PROPOSITION C.1.2. If the RHP of v(λ) := (I − w−(λ))−1(I + w+(λ)) =(χc−(λ))−1χc+(λ), λ ∈ >, where w±(λ) ∈ Hk(>±,M2(C)), admits a fundamen-

tal solution χc(λ) ∈ Hj

I (>,M2(C)) for some j ∈ Z�1, then it is unique inL2

I (>,M2(C)) := H 0I (>,M2(C)).

PROPOSITION C.1.3. If the RHP of v(λ) := (I − w−(λ))−1(I + w+(λ)) =(χc−(λ))−1χc+(λ), λ ∈ >, where w±(λ) ∈ Hk(>±,M2(C)), admits a fundamental

solution χc(λ) ∈ HjI (>,M2(C)) for some j ∈ Z�0, then id − Cw is invertible on

Hj ′I (>,M2(C)) ∀j ′ ∈ {0, 1, . . . , k}, k ∈ Z�1.

PROPOSITION C.1.4. Suppose that w±(λ) ∈ Hk(>±,M2(C)). If id − Cw isinvertible on H

jI (>,M2(C)) for any j � k, k ∈ Z�1, then it is invertible ∀j � k.

Denote the Schwarz reflection of an M2(C)-valued function by f S(λ) :=(f (λ))†, where † denotes Hermitian conjugation, and, for a subset of C, as thereflection about R.

THEOREM C.1.4. If > is a Schwarz reflection invariant contour about R, v(λ) ∈SHk

I (>−,M2(C)) ∗ SHk

I (>+,M2(C)), v(∞) = I, v(·) is positive definite on R,

Re(v(λ)) �R> 0, and v(λ) �>\R= σ−1vS(λ) �>\R σ , where σ is a constant, invert-ible, finite-order matrix involution which changes the sign(s) of some (or all) ofthe elements of the matrix on which it (and its inverse) is multiplied, then all thepartial indices of v(λ) are zero, ki = 0, i ∈ {1, 2}. In this case, the RHP for v(λ)is solvable.

The singular RHP, that is, the RH factorisation problem with isolated singu-larities (in this work, first-order poles), is now introduced. Let ζ ∈ C. For theremainder of this Appendix, the same symbol is used to denote an M2(C)-valuedfunction analytic in a punctured neighbourhood of ζ and the germ (the set ofequivalence classes of analytic continuations) at ζ it represents, with the algebra ofall such germs denoted by Aζ , and SAζ := {ϕζ (λ) ∈ Aζ ; det(ϕζ (λ)) = 1}. LetD ⊂ C, with card(D) < ∞. Set D± := D ∩�∓. Define Hk(>±∪D,M2(C)) :=


Hk(>±,M2(C)) ⊕ (⊕

ζ∈D Aζ ). An element in⊕

α∈{±}Hk(>α ∪ D,M2(C)) isrepresented either as ϕ(λ) := (ϕc(λ), ϕζ (λ))ζ∈D , or

ϕ(λ) :={ϕc(λ), λ ∈ >,ϕζ (λ), λ ≈ ζ, ζ ∈ D ,

where ϕc(λ) ∈ ⊕α∈{±}Hk(>α,M2(C)), and ϕζ (λ) ∈ Aζ (in the above, the sub-

script c is used to connote ‘continuous’, while the subscript ζ (for ζ ∈ D) isused to connote ‘discrete’). The Cauchy integral operators, C±, are defined on⊕

α∈{±}Hk(>α ∪D,M2(C)) in the following sense: construct the augmented con-

tour >aug := >∪(∪ζ∈D>ζ ), where >ζ are sufficiently small, mutually disjoint, anddisjoint with respect to >, disks oriented counter-clockwise (respectively, clock-wise) ∀ζ ∈ D+ (respectively, ∀ζ ∈ D−). Since, with the above-given conditionson >ζ , ζ ∈ D , and, for each ϕ(λ) ∈⊕α∈{±}H

k(>α ∪D,M2(C)), ϕ(λ)�λ∈>aug∃,

it represents an element in⊕

α∈{±}Hk(>α

aug,M2(C)); hence, (C±ϕ)(λ) are defined,and (C±ϕ)(λ) ∈ Hk(>±

aug,M2(C)). Hereafter, (C±ϕ)(λ) are to be understood aselements in Hk(>±∪D,M2(C)). For ζ ∈ D+, (C+ϕ)(λ) extends analytically intothe disk bounded by >ζ , and (C−ϕ)(λ) := (C+ϕ)(λ)− ϕ(λ) extends analyticallyinto the punctured disk; therefore, they represent germs in Aζ , denoted by f ±

ζ ,respectively. Similarly, for ζ ∈ D−, (C−ϕ)(λ) extends analytically into the diskbounded by >ζ , and (C+ϕ)(λ) := (C−ϕ)(λ)+ ϕ(λ) extends analytically into thepunctured disk; therefore, they represent germs in Aζ , denoted by f ∓

ζ , respec-tively. Write f ±

c := (C±ϕ)(λ) �λ∈>, and define f ± := (f ±c , f ±

ζ )ζ∈D ∈ Hk(>± ∪D,M2(C)). From the construction above, C±:

⊕α∈{±}H

k(>α ∪ D,M2(C)) →Hk(>±∪D,M2(C)), and (C±ϕ)(λ) = f ±. In this sense, C± are called the Cauchyintegral operators with singular support >∪D . The following notion of piecewise-holomorphic matrix-valued function has been used throughout this paper. For anM2(C)-valued function, $(λ), say, the ‘symbol’ $(λ) := ($c(λ),$ζ (λ))ζ∈D issaid to be a piecewise-holomorphic matrix-valued function with respect to thecontour > ∪ D if $c(λ) is a piecewise-holomorphic matrix-valued function on� \ D and $ζ(λ) ∈ Aζ is analytic at each ζ ∈ D . The boundary values $±(λ),if they exist, of the (generalised) holomorphic matrix-valued function $(λ) :=($c(λ),$ζ (λ))ζ∈D are defined by

$+(λ) := ($c(λ))+, λ ∈ >,$c(λ), λ ≈ ζ, ζ ∈ D−,$ζ (λ), λ ≈ ζ, ζ ∈ D+,

$−(λ) := ($c(λ))−, λ ∈ >,$c(λ), λ ≈ ζ, ζ ∈ D+,$ζ (λ), λ ≈ ζ, ζ ∈ D−,

(C.1)

where ($c(λ))± := lim λ′→λλ′∈�±

$c(λ′). Define H k(C \ > ∪ D,M2(C)) := {$(λ);

$±(λ)− $±(∞) ∈ ranC±}, and SH k(C \ > ∪ D,M2(C)) := {$(λ) ∈ H k(C \> ∪D,M2(C)); det($(λ)) = 1}.

410 A. H. VARTANIAN

THEOREM C.1.5. Every v(λ) ∈ SHkI (>

− ∪D,M2(C))∗SHkI (>

+ ∪D,M2(C))

admits an RH factorisation v(λ) := (χ−(λ))−1�(λ)χ+(λ), where χ(λ) ∈ SH k(C\> ∪D,M2(C)), �(λ) is defined in Theorem C.1.1, and λ± ∈ D± ∪ (�± \D∓).

THEOREM C.1.6. If > ∪ D is Schwarz reflection invariant with respect to R,v(λ) ∈ SHk

I (>− ∪D,M2(C))∗SHk

I (>+ ∪D,M2(C)), v(∞) = I, Re(v(λ))�λ∈R>

0, and v(λ)�λ∈(>∪D)\R= σ−1vS(λ)�λ∈(>∪D)\R σ , where σ is a constant, invertible,finite-order matrix involution which changes the sign(s) of some (or all) of theelements of the matrix on which it (and its inverse) is multiplied, then all the partialindices of v(λ) are zero, ki = 0, i ∈ {1, 2}. In this case, the RHP for v(λ) issolvable.

Note that, for D ≡ ∅, Theorem C.1.6 reduces to Theorem C.1.4. The asymp-totic analysis of the latter part of the above-given paradigm, related to the singularRHP (when D �≡ ∅ and card(D) < ∞), is the subject of the present asymp-totic study. Using the results of this subsection, the very important Lemma 2.4of [50], and the Deift–Zhou nonlinear steepest descent method [55], the (rigorous)asymptotic analysis, as |t| → ∞ and |x| → ∞ such that z0 := x/t ∼ O(1) and∈ R\{−2, 0, 2}, of the RHP for m(ζ) formulated in Lemma 2.1.2, for σd ≡ ∅, wascompleted in [38].

Acknowledgements

The author is very grateful to X. Zhou for the invitation to Duke University and forthe opportunity to complete this work. The author is also grateful to the refereesfor helpful suggestions.

References

1. Agrawal, G. P.: Nonlinear Fiber Optics, 2nd edn, Academic Press, San Diego, 1995.2. Kodama, Y.: The Whitham equations for optical communications: mathematical theory of NRZ,

SIAM J. Appl. Math. 59 (1999), 2162–2192.3. Weiner, A. M.: Dark optical solitons, In: J. R. Taylor (ed.), Optical Solitons – Theory and

Experiment, Cambridge Stud. Modern Optics 10, Cambridge Univ. Press, Cambridge, 1992,pp. 378–408.

4. Lundquist, P. B., Andersen, D. R. and Swartzlander, G. A., Jr.: Asymptotic behavior of the self-defocusing nonlinear Schrödinger equation for piecewise constant initial conditions, J. Opt.Soc. Amer. B 12 (1995), 698–703.

5. Lundina, D. Sh. and Marchenko, V. A.: Compactness of the set of multisoliton solutions of thenonlinear Schrödinger equation, Russian Acad. Sci. Sb. Math. 75 (1993), 429–443.

6. Boutet de Monvel, A. and Marchenko, V.: The Cauchy problem for nonlinear Schrödingerequation with bounded initial data, Mat. Fiz. Anal. Geom. 4 (2000), 3–45.

7. Novikov, S. P., Manakov, S. V., Pitaevskii, L. P. and Zakharov, V. E.: Theory of Solitons: TheInverse Scattering Method, Plenum, New York, 1984.

8. Ablowitz, M. J. and Clarkson P. A.: Solitons, Nonlinear Evolution Equations and InverseScattering, London Math. Soc. Lecture Notes Ser. 149, Cambridge Univ. Press, Cambridge,1991.


9. Cherednik, I.: Basic Methods of Soliton Theory, Adv. Ser. Math. Phys. 25, World Scientific,Singapore, 1996.

10. Schuur, P. C.: Asymptotic Analysis of Soliton Problems, Lecture Notes in Math. 1232, Springer-Verlag, Berlin, 1986.

11. Zakharov, V. E. and Manakov, S. V.: On the complete integrability of a nonlinear Schrödingerequation, Theoret. Math. Phys. 19 (1974), 551–559.

12. Faddeev, L. D. and Takhtajan, L. A.: Hamiltonian Methods in the Theory of Solitons, Springer-Verlag, Berlin, 1987.

13. Zakharov, V. E. and Shabat, A. B.: Interaction between solitons in a stable medium, SovietPhys. JETP 37 (1973), 823–828.

14. Asano, N. and Kato, Y.: Non-self-adjoint Zakharov–Shabat operator with a potential of thefinite asymptotic values. I. Direct spectral and scattering problems, J. Math. Phys. 22 (1981),2780–2793; II. Inverse problem, J. Math. Phys. 25 (1984), 570–588.

15. Kawata, T. and Inoue, H.: Inverse scattering method for the nonlinear evolution equations undernonvanishing conditions, J. Phys. Soc. Japan 44 (1978), 1722–1729.

16. Frolov, I. S.: Inverse scattering problem for a Dirac system on the whole axis, Soviet Math.Dokl. 13 (1972), 1468–1472.

17. Anders, I. A. and Kotlyarov, V. P.: Characterization of the scattering data of the Schrödingerand Dirac operators, Theoret. Math. Phys. 88 (1991), 725–734.

18. Boiti, M. and Pempinelli, F.: The spectral transform for the NLS equation with left-rightasymmetric boundary conditions, Nuovo Cimento B 69 (1982), 213–227.

19. Kawata, T. and Inoue, H.: Exact solutions of the derivative nonlinear Schrödinger equationunder the nonvanishing conditions, J. Phys. Soc. Japan 44 (1978), 1968–1976;Kawata, T., Sakai, J. and Kobayashi, N.: Inverse method for the mixed nonlinear Schrödingerequation and soliton solutions, J. Phys. Soc. Japan 48 (1980), 1371–1379.

20. Marchenko, V. A.: The Cauchy problem for the KdV equation with nondecreasing initial data,In: V. E. Zakharov (ed.), What is Integrability?, Springer Ser. Nonlinear Dynam., Springer-Verlag, Berlin, 1991, pp. 273–318.

21. Boutet de Monvel, A., Khruslov, E. Ya. and Kotlyarov, V. P.: The Cauchy problem for thesinh-Gordon equation and regular solitons, Inverse Problems 14 (1998), 1403–1427.

22. Boutet de Monvel, A., Egorova, I. and Khruslov, E.: Soliton asymptotics of the Cauchy problemsolution for the Toda lattice, Inverse Problems 13 (1997), 223–237;Boutet de Monvel, A. and Egorova, I.: The Toda lattice with step-like initial data. Solitonasymptotics, Inverse Problems 16 (2000), 955–977.

23. Kotlyarov, V. P. and Khruslov, E. Ya.: Solitons of the nonlinear Schrödinger equation generatedby the continuum, Theoret. Math. Phys. 68 (1986), 751–761; Asymptotic solitons of the modi-fied Korteweg–de Vries equation, Inverse Problems 5 (1989), 1075–1088;Kotlyarov, V. P.: Asymptotic solitons of the sine–Gordon equation, Theoret. Math. Phys. 80(1989), 679–689.

24. Khruslov, E. Ya.: Asymptotic solution of the Cauchy problem for the Korteweg–de Vriesequation with step-type initial data, Math. USSR-Sb. 99 (1976), 261–281 (in Russian).

25. Kirsch, W. and Kotlyarov, V.: Soliton asymptotics of solutions of the sine–Gordon equation,Math. Phys. Anal. Geom. 2 (1999), 25–51.

26. Khruslov, E. Ya. and Stephan, H.: Splitting of some nonlocalized solutions of the Korteweg–deVries equation into solitons, Mat. Fiz. Anal. Geom. 5 (1998), 49–67.

27. Borisov, A. B. and Kiseliev, V. V.: Inverse problem for an elliptic sine-Gordon equation withan asymptotic behaviour of the cnoidal-wave type, Inverse Problems 5 (1989), 959–982.

28. Vekslerchik, V. E. and Konotop, V. V.: Discrete nonlinear Schrödinger equation undernonvanishing boundary conditions, Inverse Problems 8 (1992), 889–909.

29. Vekslerchik, V. E.: Inverse scattering transform for the O(3, 1) nonlinear σ -model, InverseProblems 12 (1996), 517–534.

412 A. H. VARTANIAN

30. Cohen, A. and Kappeler, T.: Scattering and inverse scattering for steplike potentials in theSchrödinger equation, Indiana Univ. Math. J. 34 (1985), 127–180.

31. Grebert, B.: Inverse scattering for the Dirac operator on the real line, Inverse Problems 8 (1992),787–807.

32. Bikbaev, R. F. and Sharipov, R. A.: Asymptotics as t → ∞ of the solution to the Cauchyproblem for the Korteweg–de Vries equation in the class of potentials with finite-gap behavioras x →±∞, Theor. Math. Phys. 78 (1989), 244–252.

33. Zakharov, V. E. and Shabat, A. B.: Integration of the nonlinear equations of mathematicalphysics by the method of the inverse scattering transform. II, Functional Anal. Appl. 13 (1980),166–173.

34. Deift, P. A., Kamvissis, S., Kriecherbauer, T. and Zhou, X.: The Toda rarefaction problem,Comm. Pure Appl. Math. 49 (1996), 35–83.

35. Gesztesy, F. and Svirsky, R.: (m)KdV Solitons on the Background of Quasi-Periodic Finite-GapSolutions, Mem. Amer. Math. Soc. 118, Amer. Math. Soc., Providence, 1995.

36. Renger, W.: Toda soliton limits on general backgrounds, J. Differential Equations 151 (1999),191–230.

37. Kitaev, A. V. and Vartanian, A. H.: Asymptotics of solutions to the modified nonlinearSchrödinger equation: Solitons on a nonvanishing continuous background, SIAM J. Math. Anal.30 (1999), 787–832.

38. Vartanian, A. H.: Long-time asymptotics of solutions to the Cauchy problem for the defocus-ing nonlinear Schrödinger equation with finite-density initial data. I. Solitonless sector, 2001arXiv:nlin.SI/0110024.

39. Its, A. R.: Asymptotics of solutions of the nonlinear Schrödinger equation and isomonodromicdeformations of systems of linear differential equations, Soviet Math. Dokl. 24 (1981), 452–456.

40. Clancey, K. and Gohberg, I.: Factorization of Matrix Functions and Singular Integral Opera-tors, Oper. Theory Adv. Appl. 3, Birkhäuser, Basel, 1981.

41. Beals, R. and Coifman, R. R.: Scattering and inverse scattering for first order systems, Comm.Pure Appl. Math. 37 (1984), 39–90.

42. Deift, P.: Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach,Courant Lecture Notes in Math. 3, CIMS, New York, 1999.

43. Fokas, A. S.: On the integrability of linear and nonlinear partial differential equations, J. Math.Phys. 41 (2000), 4188–4237.

44. Zhou, X.: The Riemann–Hilbert problem and inverse scattering, SIAM J. Math. Anal. 20 (1989),966–986.

45. Zhou, X.: Direct and inverse scattering transforms with arbitrary spectral singularities, Comm.Pure Appl. Math. 42 (1989), 895–938.

46. Zhou, X.: Inverse scattering transform for systems with rational spectral dependence, J. Differ-ential Equations 115 (1995), 277–303.

47. Zhou, X.: Strong regularizing effect of integrable systems, Comm. Partial Differential Equa-tions 22 (1997), 503–526.

48. Its, A. R. and Ustinov, A. F.: The time asymptotics of the solution of the Cauchy problem forthe nonlinear Schrödinger equation with finite density boundary conditions, Dokl. Akad. NaukSSSR 291 (1986), 91–95 (in Russian).

49. Its, A. R. and Ustinov, A. F.: Formulation of scattering theory for the nonlinear Schrödingerequation with boundary conditions of the finite density type in a soliton-free sector, J. SovietMath. 54 (1991), 900–905.

50. Zhou, X.: L2-Sobolev space bijectivity of the scattering and inverse scattering transforms,Comm. Pure Appl. Math. 51 (1998), 697–731.

51. Gradshteyn, I. S. and Ryzhik, I. M.: Tables of Integrals, Series, and Products, 5th edn, A. Jeffrey(ed.), Academic Press, San Diego, 1994.


52. Zakharov, V. E. and Shabat, A. B.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Phys. JETP 34 (1972), 62–69.

53. Deift, P. and Zhou, X.: Direct and inverse scattering on the line with arbitrary singularities,Comm. Pure Appl. Math. 44 (1991), 485–533.

54. Deift, P. and Zhou, X.: Long-time asymptotics for integrable systems. Higher order theory,Comm. Math. Phys. 165 (1994), 175–191.

55. Deift, P. and Zhou, X.: A steepest descent method for oscillatory Riemann–Hilbert problems.Asymptotics for the MKdV equation, Ann. of Math. 137 (1993), 295–368.

56. Vartanian, A. H.: Higher order asymptotics of the modified nonlinear Schrödinger equation,Comm. Partial Differential Equations 25 (2000), 1043–1098.

Mathematical Physics, Analysis and Geometry 5: 415–416, 2002. 415

Contents of Volume 5 (2002)

Volume 5 No. 1 2002

N. N. KHURI / Inverse Scattering, the Coupling Constant Spectrum,and the Riemann Hypothesis 1–63

M. BEN CHROUDA and H. OUERDIANE / Algebras of Operators onHolomorphic Functions and Applications 65–76

MICHEL TALAGRAND / On the Gaussian Perceptron at High Tem-perature 77–99

Volume 5 No. 2 2002

DANIEL BUMP, PERSI DIACONIS and JOSEPH B. KELLER /Unitary Correlations and the Fejér Kernel 101–123

ROSSELLA BARTOLO and ANNA GERMINARIO / TrajectoriesJoining Two Submanifolds under the Action of Gravitationaland Electromagnetic Fields on Static Spacetimes 125–143

LECH ZIELINSKI / Asymptotic Distribution of Eigenvalues for a Classof Second-Order Elliptic Operators with Irregular Coefficientsin R

d 145–182

F. ALBERTO GRÜNBAUM and PLAMEN ILIEV / Heat Kernel Ex-pansions on the Integers 183–200

Volume 5 No. 3 2002

G. RUDOLPH, M. SCHMIDT and I. P. VOLOBUEV / Classification ofGauge Orbit Types for SU(n)-Gauge Theories 201–241

PAVEL KURASOV and SERGUEI NABOKO / On the Essential Spec-trum of a Class of Singular Matrix Differential Operators.I: Quasiregularity Conditions and Essential Self-adjointness 243–286

BERNHARD G. BODMANN / A Construction of Berezin–ToeplitzOperators via Schrödinger Operators and the Probabilistic Rep-resentation of Berezin–Toeplitz Semigroups Based on PlanarBrownian Motion 287–306

416 CONTENTS OF VOLUME 5

Volume 5 No. 4 2002

M. F. BORGES / Geometrical Lagrangian for a Supersymmetric Yang–Mills Theory on the Group Manifold 307–318

A. H. VARTANIAN / Long-Time Asymptotics of Solutions to theCauchy Problem for the Defocusing Nonlinear SchrödingerEquation with Finite-Density Initial Data. II. Dark Solitons onContinua 319–413

mathematical physics, analysis and geometry - volume 5

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